G.187 magnitudes ratio equal

V.9

Magnitudes which have the same ratio to the same are equal; and magnitudes to which the same has the same ratio are equal.
magnitudes_ratio_equal()
prove A = B; A = B
A B C
assert Let each of the magnitudes A, B have the same ratio to C
∵ unequal_magnitudes_greaterassert Otherwise, each of the magnitudes A, B would not have had the same ratio to C
assert But it has
assert A = B
assert Let C have the same ratio to each of the magnitudes A, B
∵ unequal_magnitudes_greaterassert Otherwise, C would not have had the same ratio to each of the magnitudes A, B
assert But it has
∴ A = B

Dependency Graph

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Required for

G.211, G.212, G.213, G.214, G.215, G.216, G.218, G.219, G.220, G.221, G.222, G.223, G.225, G.226, G.227, G.228, G.229, G.231, G.232, G.234, G.235, G.236, G.237, G.238, G.239, G.240, G.241, G.243, G.244, G.245, G.273, G.278, G.279, G.280, G.281, G.285, G.289, G.290, G.292, G.295, G.297, G.298, G.299, G.301, G.303, G.304, G.305, G.306, G.314, G.315, G.316, G.317, G.322, G.323, G.324, G.325, G.326, G.327, G.328, G.329, G.330, G.331, G.332, G.333, G.334, G.335, G.337, G.338, G.339, G.342, G.343, G.344, G.345, G.348, G.349, G.364, G.370, G.377, G.378, G.381, G.382, G.383, G.384, G.385, G.386, G.389, G.390, G.393, G.394, G.395, G.396, G.397, G.398, G.399, G.400, G.401, G.402, G.403, G.404, G.405, G.406, G.407, G.408, G.409, G.410, G.411, G.412, G.413, G.414, G.415, G.416, G.417, G.418, G.419, G.420, G.421, G.423, G.425, G.427, G.429, G.431, G.433, G.434, G.435, G.436, G.437, G.438, G.439, G.440, G.441, G.442, G.443, G.444, G.445, G.446, G.447, G.448, G.449, G.450, G.451, G.452, G.453, G.457, G.458, G.459, G.460, G.461, G.462, G.463, G.464, G.465, G.466, G.467, G.468, G.470, G.472, G.474, G.476, G.477, G.478, G.479, G.480, G.481, G.482, G.483, G.484, G.485, G.486, G.487, G.488, G.489, G.490, G.491, G.492, G.493, G.494, G.495, G.496, G.497, G.498, G.499, G.500, G.501, G.520, G.528, G.536, G.541, G.542, G.543, G.544, G.546, G.547, G.550, G.551, G.552, G.554, G.555, G.556, G.557, G.558, G.559, G.560, G.563, G.565, G.570, G.571, G.572, G.573, G.577, G.578, G.579, G.582, G.583, G.584, G.586, G.587, G.588, G.589, G.592, G.597, G.600, G.601