let D be compact with_the_max_arc Subset of (TOP-REAL 2); :: thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in D implies LSeg ((LMP D),|[0,(- 3)]|) c= south_halfline (LMP D) )
set p = LMP D;
assume A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in D ; :: thesis: LSeg ((LMP D),|[0,(- 3)]|) c= south_halfline (LMP D)
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in LSeg ((LMP D),|[0,(- 3)]|) or x in south_halfline (LMP D) )
assume A2: x in LSeg ((LMP D),|[0,(- 3)]|) ; :: thesis: x in south_halfline (LMP D)
then reconsider x = x as Point of (TOP-REAL 2) ;
A3: LMP D in LSeg ((LMP D),|[0,(- 3)]|) by RLTOPSP1:68;
A4: LSeg ((LMP D),|[0,(- 3)]|) is vertical by A1, Th82;
then A5: x `1 = (LMP D) `1 by A2, A3;
A6: |[0,(- 3)]| = |[(|[0,(- 3)]| `1),(|[0,(- 3)]| `2)]| by EUCLID:53;
A7: LMP D = |[((LMP D) `1),((LMP D) `2)]| by EUCLID:53;
|[0,(- 3)]| in LSeg ((LMP D),|[0,(- 3)]|) by RLTOPSP1:68;
then A8: |[0,(- 3)]| `1 = (LMP D) `1 by A3, A4;
|[0,(- 3)]| `2 <= (LMP D) `2 by A1, Lm23, Th84, JORDAN21:31;
then x `2 <= (LMP D) `2 by A2, A6, A7, A8, JGRAPH_6:1;
hence x in south_halfline (LMP D) by A5, TOPREAL1:def 12; :: thesis: verum