let p be Point of (TOP-REAL 2); :: thesis: for D being compact with_the_max_arc Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in D & p in LSeg (|[0,3]|,(UMP D)) holds
(UMP D) `2 <= p `2
let D be compact with_the_max_arc Subset of (TOP-REAL 2); :: thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in D & p in LSeg (|[0,3]|,(UMP D)) implies (UMP D) `2 <= p `2 )
set x = UMP D;
assume that
A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in D and
A2: p in LSeg (|[0,3]|,(UMP D)) ; :: thesis: (UMP D) `2 <= p `2
A3: UMP D in LSeg (|[0,3]|,(UMP D)) by RLTOPSP1:68;
A4: LSeg (|[0,3]|,(UMP D)) is vertical by A1, Th81;
A5: |[0,3]| = |[(|[0,3]| `1),(|[0,3]| `2)]| by EUCLID:53;
A6: UMP D = |[((UMP D) `1),((UMP D) `2)]| by EUCLID:53;
|[0,3]| in LSeg (|[0,3]|,(UMP D)) by RLTOPSP1:68;
then A7: |[0,3]| `1 = (UMP D) `1 by A3, A4;
(UMP D) `2 <= |[0,3]| `2 by A1, Lm21, Th83, JORDAN21:30;
hence (UMP D) `2 <= p `2 by A2, A5, A6, A7, JGRAPH_6:1; :: thesis: verum