In III.7 Euclid proves that $FA > FB > FC > FG > FE$ using the Triangle Inequality and Hinge Theorem. He then goes on to prove using triangles that if $P$ is a point on the side of the diameter containing $G$, then $FP \neq FG$.
Why does he need to prove the second claim (i.e. that no segment on the same side has equal length to $FG$) via triangles? If the length is strictly decreasing as we sweep from $FA$ to $FE$, doesn't that make each length unique?