G.291 numbers measure number

VII.35

If two numbers measure any number, the least number measured by them will also measure the same.
numbers_measure_number()
prove E also measures / C D /
A B C F D E
assert Let the two numbers A, B measure any number / C D /, and let E be the least that they measure
assert If E does not measure / C D /, let E, measuring / F D /, leave / C F / less than itself
assert A, B measure E, and E measures / F D /,
assert A, B will also measure / F D /
assert They also measure / C D /
assert they will also measure / C F / which < E ⇒⇐
∴ E cannot fail to measure / C D /; therefore it measures it

Dependency Graph

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Required for

G.292, G.295, G.299, G.301, G.384, G.385, G.389, G.390, G.396, G.397, G.398, G.399, G.400, G.401, G.402, 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.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.584, G.589, G.597, G.600, G.601