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 (LMP D),|[0 ,(- 3)]| holds
p `2 <= (LMP D) `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 (LMP D),|[0 ,(- 3)]| implies p `2 <= (LMP D) `2 )
set x = LMP D;
assume that
A1: |[(- 1),0 ]|,|[1,0 ]| realize-max-dist-in D and
A2: p in LSeg (LMP D),|[0 ,(- 3)]| ; :: thesis: p `2 <= (LMP D) `2
A3: LMP D in LSeg (LMP D),|[0 ,(- 3)]| by RLTOPSP1:69;
A4: LSeg (LMP D),|[0 ,(- 3)]| is vertical by A1, Th82;
A5: |[0 ,(- 3)]| = |[(|[0 ,(- 3)]| `1 ),(|[0 ,(- 3)]| `2 )]| by EUCLID:57;
A6: LMP D = |[((LMP D) `1 ),((LMP D) `2 )]| by EUCLID:57;
|[0 ,(- 3)]| in LSeg (LMP D),|[0 ,(- 3)]| by RLTOPSP1:69;
then A7: |[0 ,(- 3)]| `1 = (LMP D) `1 by A3, A4, SPPOL_1:def 3;
|[0 ,(- 3)]| `2 <= (LMP D) `2 by A1, Lm23, Th84, JORDAN21:44;
hence p `2 <= (LMP D) `2 by A2, A5, A6, A7, JGRAPH_6:9; :: thesis: verum