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    "markdown": "\\iint_{D} n^2 \\leq y^2 \\, dx \\, dy\n\\mathbb{E}[i] = \\int x f_i(x) dx\n## ملاحظة\nدُونِ مَسَافَةِ الْقَصْرِ لِحَاجَةٍ كَتَطَهُّرٍ أَوْ نَوَى الرُّجُوعَ لَهُ وَهُوَ مُسْتَقِلٌّ مَاكِثٌ وَلَوْ بِمَكَانٍ لَا يَصْلُحُ لِلْإِقَامَةِ، فَإِنْ كَانَ وَطَنَهُ صَارَ مُقِيمًا بِابْتِدَاءِ رُجُوعِهِ أَوْ نِيَّتِهِ فَلَا يَتَرَخَّصُ فِي إقَامَتِهِ\n\\sigma_{n+1} = \\sigma_n - \\alpha \\nabla f(\\sigma_n)\n## نظرية\nوَلَا رُجُوعِهِ إلَى أَنْ يُفَارِقَ وَطَنَهُ تَغْلِيبًا لِلْوَطَنِ، وَحَكَى فِيهِ فِي أَصْلِ الرَّوْضَةِ وَجْهًا شَاذًّا أَنَّهُ يَتَرَخَّصُ إلَى أَنْ يَصِلَهُ اهـ. وَالْأَوَّلُ هُوَ الْمُعْتَمَدُ وَإِنْ نَازَعَ فِيهِ الْبُلْقِينِيُّ وَالْأَذْرَعِيُّ وَغَيْرُهُمَا وَإِنْ لَمْ يَكُنْ وَطَنَهُ تَرَخَّصَ وَإِنْ دَخَلَهُ وَلَوْ كَانَ دَارَ إقَامَتِهِ لِانْتِفَاءِ الْوَطَنِ فَكَانَتْ كَسَائِرِ الْمَنَازِلِ. فَإِنْ رَجَعَ مِنْ السَّفَرِ\n\\lambda_{n+1} = \\lambda_n - \\alpha \\nabla f(\\lambda_n)\n## برهان\nالطَّوِيلِ (انْتَهَى سَفَرُهُ بِبُلُوغِهِ مَا شُرِطَ مُجَاوَزَتُهُ ابْتِدَاءً) مِنْ سُورٍ أَوْ غَيْرِهِ فَيَتَرَخَّصُ إلَى أَنْ يَصِلَ إلَى ذَلِكَ. فَإِنْ قِيلَ: يَنْبَغِي أَنْ لَا يَنْتَهِيَ سَفَرُهُ إلَّا بِدُخُولِ الْعُمْرَانِ أَوْ السُّورِ كَمَا لَا يَصِيرُ مُسَافِرًا إلَّا بِخُرُوجِهِ\n\\nabla_{y} J(y) = \\frac{1}{m} X^T ( \\sigma(X y) - Y )\n## نظرية\nمِنْهُ، وَفِي نُسْخَةٍ مِنْ الرَّوْضِ مَا يَدُلُّ لِذَلِكَ. أُجِيبَ بِأَنَّ مَا فِي الْمَتْنِ هُوَ الْمَنْقُولُ، وَالْفَرْقُ أَنَّ الْأَصْلَ الْإِقَامَةُ فَلَا تَنْقَطِعُ إلَّا بِتَحَقُّقِ السَّفَرِ وَتَحَقُّقُهُ بِخُرُوجِهِ مِنْ ذَلِكَ وَالسَّفَرُ عَلَى خِلَافِ الْأَصْلِ فَانْقَطَعَ بِمُجَرَّدِ الْوُصُولِ وَإِنْ لَمْ يَدْخُلْ، فَعُلِمَ أَنَّهُ يَنْتَهِي بِمُجَرَّدِ بُلُوغِهِ مُبْتَدَأَ سَفَرِهِ مِنْ وَطَنِهِ وَإِنْ كَانَ\n\\mathbb{E}[j] = \\int x f_j(x) dx + \\frac{i}{\\mu^2}\n## مبرهنة\nمَارًّا بِهِ فِي سَفَرِهِ كَأَنْ خَرَجَ مِنْهُ ثُمَّ رَجَعَ مِنْ بَعِيدٍ قَاصِدًا الْمُرُورَ بِهِ مِنْ غَيْرِ إقَامَةٍ لَا مِنْ بَلَدٍ يَقْصِدُهُ وَلَا بَلَدٍ لَهُ فِيهَا أَهْلٌ وَعَشِيرَةٌ لَمْ يَنْوِ الْإِقَامَةَ بِكُلٍّ مِنْهُمَا فَلَا يَنْتَهِي سَفَرُهُ بِوُصُولِهِ إلَيْهِمَا بِخِلَافِ مَا إذَا نَوَى الْإِقَامَةَ بِهِمَا فَإِنَّهُ يَنْتَهِي سَفَرُهُ بِذَلِكَ\n\\vec{F} = m\\vec{a}",
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                "text": "إلَّا",
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            {
                "text": "بِدُخُولِ",
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            {
                "text": "الْعُمْرَانِ",
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            {
                "text": "أَوْ",
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                "text": "السُّورِ",
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                "text": "كَمَا",
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                "text": "لَا",
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                "text": "يَصِيرُ",
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                "text": "مُسَافِرًا",
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            {
                "text": "إلَّا",
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            {
                "text": "بِخُرُوجِهِ",
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            {
                "text": "\\nabla_{y}",
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            {
                "text": "J(y)",
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            {
                "text": "=",
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            {
                "text": "\\frac{1}{m}",
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                "text": "X^T",
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                "text": "(",
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            {
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            {
                "text": "-",
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            {
                "text": "Y",
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            {
                "text": ")",
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            {
                "text": "##",
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            },
            {
                "text": "نظرية",
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            },
            {
                "text": "مِنْهُ،",
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                "start_index": 1247
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            {
                "text": "وَفِي",
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            },
            {
                "text": "نُسْخَةٍ",
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            {
                "text": "مِنْ",
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                "text": "الرَّوْضِ",
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            {
                "text": "مَا",
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            {
                "text": "يَدُلُّ",
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                "text": "لِذَلِكَ.",
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            {
                "text": "أُجِيبَ",
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                "text": "بِأَنَّ",
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                "text": "مَا",
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            {
                "text": "فِي",
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            },
            {
                "text": "الْمَتْنِ",
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            {
                "text": "هُوَ",
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                "text": "الْمَنْقُولُ،",
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            {
                "text": "وَالْفَرْقُ",
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            {
                "text": "أَنَّ",
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                "start_index": 1372
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            {
                "text": "الْأَصْلَ",
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                "start_index": 1378
            },
            {
                "text": "الْإِقَامَةُ",
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                "start_index": 1388
            },
            {
                "text": "فَلَا",
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                "start_index": 1401
            },
            {
                "text": "تَنْقَطِعُ",
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            {
                "text": "إلَّا",
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            },
            {
                "text": "بِتَحَقُّقِ",
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                "start_index": 1424
            },
            {
                "text": "السَّفَرِ",
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                "start_index": 1436
            },
            {
                "text": "وَتَحَقُّقُهُ",
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            },
            {
                "text": "بِخُرُوجِهِ",
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            },
            {
                "text": "مِنْ",
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            },
            {
                "text": "ذَلِكَ",
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            },
            {
                "text": "وَالسَّفَرُ",
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            },
            {
                "text": "عَلَى",
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            },
            {
                "text": "خِلَافِ",
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                "start_index": 1502
            },
            {
                "text": "الْأَصْلِ",
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                "start_index": 1510
            },
            {
                "text": "فَانْقَطَعَ",
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            },
            {
                "text": "بِمُجَرَّدِ",
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            },
            {
                "text": "الْوُصُولِ",
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            },
            {
                "text": "وَإِنْ",
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            },
            {
                "text": "لَمْ",
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            },
            {
                "text": "يَدْخُلْ،",
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            },
            {
                "text": "فَعُلِمَ",
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                "text": "أَنَّهُ",
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                "text": "يَنْتَهِي",
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            {
                "text": "بِمُجَرَّدِ",
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                "text": "بُلُوغِهِ",
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            {
                "text": "مُبْتَدَأَ",
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            {
                "text": "سَفَرِهِ",
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            {
                "text": "مِنْ",
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            },
            {
                "text": "وَطَنِهِ",
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            },
            {
                "text": "وَإِنْ",
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            },
            {
                "text": "كَانَ",
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            },
            {
                "text": "\\mathbb{E}[j]",
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                "start_index": 1673
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            {
                "text": "=",
                "confidence": 1.0,
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            {
                "text": "\\int",
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            {
                "text": "x",
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            {
                "text": "f_j(x)",
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            {
                "text": "dx",
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            {
                "text": "+",
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            {
                "text": "\\frac{i}{\\mu^2}",
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            {
                "text": "##",
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            {
                "text": "مبرهنة",
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            {
                "text": "مَارًّا",
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            {
                "text": "بِهِ",
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                "text": "فِي",
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                "text": "سَفَرِهِ",
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                "text": "كَأَنْ",
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            {
                "text": "خَرَجَ",
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            {
                "text": "مِنْهُ",
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            {
                "text": "ثُمَّ",
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            },
            {
                "text": "رَجَعَ",
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                "start_index": 1787
            },
            {
                "text": "مِنْ",
                "confidence": 1.0,
                "start_index": 1794
            },
            {
                "text": "بَعِيدٍ",
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                "start_index": 1799
            },
            {
                "text": "قَاصِدًا",
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                "start_index": 1807
            },
            {
                "text": "الْمُرُورَ",
                "confidence": 1.0,
                "start_index": 1816
            },
            {
                "text": "بِهِ",
                "confidence": 1.0,
                "start_index": 1827
            },
            {
                "text": "مِنْ",
                "confidence": 1.0,
                "start_index": 1832
            },
            {
                "text": "غَيْرِ",
                "confidence": 1.0,
                "start_index": 1837
            },
            {
                "text": "إقَامَةٍ",
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                "start_index": 1844
            },
            {
                "text": "لَا",
                "confidence": 1.0,
                "start_index": 1853
            },
            {
                "text": "مِنْ",
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                "start_index": 1857
            },
            {
                "text": "بَلَدٍ",
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                "start_index": 1862
            },
            {
                "text": "يَقْصِدُهُ",
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                "start_index": 1869
            },
            {
                "text": "وَلَا",
                "confidence": 1.0,
                "start_index": 1880
            },
            {
                "text": "بَلَدٍ",
                "confidence": 1.0,
                "start_index": 1886
            },
            {
                "text": "لَهُ",
                "confidence": 1.0,
                "start_index": 1893
            },
            {
                "text": "فِيهَا",
                "confidence": 1.0,
                "start_index": 1898
            },
            {
                "text": "أَهْلٌ",
                "confidence": 1.0,
                "start_index": 1905
            },
            {
                "text": "وَعَشِيرَةٌ",
                "confidence": 1.0,
                "start_index": 1912
            },
            {
                "text": "لَمْ",
                "confidence": 1.0,
                "start_index": 1924
            },
            {
                "text": "يَنْوِ",
                "confidence": 1.0,
                "start_index": 1929
            },
            {
                "text": "الْإِقَامَةَ",
                "confidence": 1.0,
                "start_index": 1936
            },
            {
                "text": "بِكُلٍّ",
                "confidence": 1.0,
                "start_index": 1949
            },
            {
                "text": "مِنْهُمَا",
                "confidence": 1.0,
                "start_index": 1957
            },
            {
                "text": "فَلَا",
                "confidence": 1.0,
                "start_index": 1967
            },
            {
                "text": "يَنْتَهِي",
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                "start_index": 1973
            },
            {
                "text": "سَفَرُهُ",
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                "start_index": 1983
            },
            {
                "text": "بِوُصُولِهِ",
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            },
            {
                "text": "إلَيْهِمَا",
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            },
            {
                "text": "بِخِلَافِ",
                "confidence": 1.0,
                "start_index": 2015
            },
            {
                "text": "مَا",
                "confidence": 1.0,
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            },
            {
                "text": "إذَا",
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            },
            {
                "text": "نَوَى",
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            },
            {
                "text": "الْإِقَامَةَ",
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                "start_index": 2040
            },
            {
                "text": "بِهِمَا",
                "confidence": 1.0,
                "start_index": 2053
            },
            {
                "text": "فَإِنَّهُ",
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                "start_index": 2061
            },
            {
                "text": "يَنْتَهِي",
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                "start_index": 2071
            },
            {
                "text": "سَفَرُهُ",
                "confidence": 1.0,
                "start_index": 2081
            },
            {
                "text": "بِذَلِكَ",
                "confidence": 1.0,
                "start_index": 2090
            },
            {
                "text": "\\vec{F}",
                "confidence": 1.0,
                "start_index": 2099
            },
            {
                "text": "=",
                "confidence": 1.0,
                "start_index": 2107
            },
            {
                "text": "m\\vec{a}",
                "confidence": 1.0,
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        {
            "top_left_x": 40,
            "top_left_y": 46,
            "bottom_right_x": 696,
            "bottom_right_y": 101,
            "content": "\\iint_{D} n^2 \\leq y^2 \\, dx \\, dy",
            "type": "text",
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            "reading_index": 1
        },
        {
            "top_left_x": 40,
            "top_left_y": 107,
            "bottom_right_x": 696,
            "bottom_right_y": 161,
            "content": "\\mathbb{E}[i] = \\int x f_i(x) dx",
            "type": "text",
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            "reading_index": 2
        },
        {
            "top_left_x": 52,
            "top_left_y": 179,
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            "content": "ملاحظة",
            "type": "title",
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        {
            "top_left_x": 52,
            "top_left_y": 205,
            "bottom_right_x": 680,
            "bottom_right_y": 250,
            "content": "دُونِ مَسَافَةِ الْقَصْرِ لِحَاجَةٍ كَتَطَهُّرٍ أَوْ نَوَى الرُّجُوعَ لَهُ وَهُوَ مُسْتَقِلٌّ مَاكِثٌ وَلَوْ بِمَكَانٍ لَا يَصْلُحُ لِلْإِقَامَةِ، فَإِنْ كَانَ وَطَنَهُ صَارَ مُقِيمًا بِابْتِدَاءِ رُجُوعِهِ أَوْ نِيَّتِهِ فَلَا يَتَرَخَّصُ فِي إقَامَتِهِ",
            "type": "text",
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        },
        {
            "top_left_x": 52,
            "top_left_y": 256,
            "bottom_right_x": 680,
            "bottom_right_y": 294,
            "content": "\\sigma_{n+1} = \\sigma_n - \\alpha \\nabla f(\\sigma_n)",
            "type": "text",
            "confidence_scores": null,
            "reading_index": 5
        },
        {
            "top_left_x": 52,
            "top_left_y": 338,
            "bottom_right_x": 680,
            "bottom_right_y": 359,
            "content": "نظرية",
            "type": "title",
            "confidence_scores": null,
            "reading_index": 6
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        {
            "top_left_x": 52,
            "top_left_y": 364,
            "bottom_right_x": 680,
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            "content": "وَلَا رُجُوعِهِ إلَى أَنْ يُفَارِقَ وَطَنَهُ تَغْلِيبًا لِلْوَطَنِ، وَحَكَى فِيهِ فِي أَصْلِ الرَّوْضَةِ وَجْهًا شَاذًّا أَنَّهُ يَتَرَخَّصُ إلَى أَنْ يَصِلَهُ اهـ. وَالْأَوَّلُ هُوَ الْمُعْتَمَدُ وَإِنْ نَازَعَ فِيهِ الْبُلْقِينِيُّ وَالْأَذْرَعِيُّ وَغَيْرُهُمَا وَإِنْ لَمْ يَكُنْ وَطَنَهُ تَرَخَّصَ وَإِنْ دَخَلَهُ وَلَوْ كَانَ دَارَ إقَامَتِهِ لِانْتِفَاءِ الْوَطَنِ فَكَانَتْ كَسَائِرِ الْمَنَازِلِ. فَإِنْ رَجَعَ مِنْ السَّفَرِ",
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        },
        {
            "top_left_x": 52,
            "top_left_y": 437,
            "bottom_right_x": 680,
            "bottom_right_y": 475,
            "content": "\\lambda_{n+1} = \\lambda_n - \\alpha \\nabla f(\\lambda_n)",
            "type": "text",
            "confidence_scores": null,
            "reading_index": 8
        },
        {
            "top_left_x": 52,
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            "bottom_right_y": 540,
            "content": "برهان",
            "type": "title",
            "confidence_scores": null,
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        },
        {
            "top_left_x": 52,
            "top_left_y": 545,
            "bottom_right_x": 680,
            "bottom_right_y": 590,
            "content": "الطَّوِيلِ (انْتَهَى سَفَرُهُ بِبُلُوغِهِ مَا شُرِطَ مُجَاوَزَتُهُ ابْتِدَاءً) مِنْ سُورٍ أَوْ غَيْرِهِ فَيَتَرَخَّصُ إلَى أَنْ يَصِلَ إلَى ذَلِكَ. فَإِنْ قِيلَ: يَنْبَغِي أَنْ لَا يَنْتَهِيَ سَفَرُهُ إلَّا بِدُخُولِ الْعُمْرَانِ أَوْ السُّورِ كَمَا لَا يَصِيرُ مُسَافِرًا إلَّا بِخُرُوجِهِ",
            "type": "text",
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        {
            "top_left_x": 52,
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            "bottom_right_x": 680,
            "bottom_right_y": 646,
            "content": "\\nabla_{y} J(y) = \\frac{1}{m} X^T ( \\sigma(X y) - Y )",
            "type": "text",
            "confidence_scores": null,
            "reading_index": 11
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        {
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            "content": "نظرية",
            "type": "title",
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            "top_left_y": 716,
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            "bottom_right_y": 784,
            "content": "مِنْهُ، وَفِي نُسْخَةٍ مِنْ الرَّوْضِ مَا يَدُلُّ لِذَلِكَ. أُجِيبَ بِأَنَّ مَا فِي الْمَتْنِ هُوَ الْمَنْقُولُ، وَالْفَرْقُ أَنَّ الْأَصْلَ الْإِقَامَةُ فَلَا تَنْقَطِعُ إلَّا بِتَحَقُّقِ السَّفَرِ وَتَحَقُّقُهُ بِخُرُوجِهِ مِنْ ذَلِكَ وَالسَّفَرُ عَلَى خِلَافِ الْأَصْلِ فَانْقَطَعَ بِمُجَرَّدِ الْوُصُولِ وَإِنْ لَمْ يَدْخُلْ، فَعُلِمَ أَنَّهُ يَنْتَهِي بِمُجَرَّدِ بُلُوغِهِ مُبْتَدَأَ سَفَرِهِ مِنْ وَطَنِهِ وَإِنْ كَانَ",
            "type": "text",
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            "reading_index": 13
        },
        {
            "top_left_x": 52,
            "top_left_y": 790,
            "bottom_right_x": 680,
            "bottom_right_y": 844,
            "content": "\\mathbb{E}[j] = \\int x f_j(x) dx + \\frac{i}{\\mu^2}",
            "type": "text",
            "confidence_scores": null,
            "reading_index": 14
        },
        {
            "top_left_x": 52,
            "top_left_y": 888,
            "bottom_right_x": 680,
            "bottom_right_y": 909,
            "content": "مبرهنة",
            "type": "title",
            "confidence_scores": null,
            "reading_index": 15
        },
        {
            "top_left_x": 52,
            "top_left_y": 914,
            "bottom_right_x": 680,
            "bottom_right_y": 958,
            "content": "مَارًّا بِهِ فِي سَفَرِهِ كَأَنْ خَرَجَ مِنْهُ ثُمَّ رَجَعَ مِنْ بَعِيدٍ قَاصِدًا الْمُرُورَ بِهِ مِنْ غَيْرِ إقَامَةٍ لَا مِنْ بَلَدٍ يَقْصِدُهُ وَلَا بَلَدٍ لَهُ فِيهَا أَهْلٌ وَعَشِيرَةٌ لَمْ يَنْوِ الْإِقَامَةَ بِكُلٍّ مِنْهُمَا فَلَا يَنْتَهِي سَفَرُهُ بِوُصُولِهِ إلَيْهِمَا بِخِلَافِ مَا إذَا نَوَى الْإِقَامَةَ بِهِمَا فَإِنَّهُ يَنْتَهِي سَفَرُهُ بِذَلِكَ",
            "type": "text",
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            "reading_index": 16
        },
        {
            "top_left_x": 52,
            "top_left_y": 964,
            "bottom_right_x": 680,
            "bottom_right_y": 1003,
            "content": "\\vec{F} = m\\vec{a}",
            "type": "text",
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            "reading_index": 17
        }
    ],
    "meta": {
        "template": "math_proof",
        "hybrid": null,
        "page_font": "interface-trial-light.ttf",
        "title_font": "interface-trial-light.ttf",
        "language": "ar",
        "script": "arabic",
        "direction": "rtl",
        "augmentation": null
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}