let P be Subset of (TOP-REAL 2); :: thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in P implies E-bound P = 1 )
assume A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in P ; :: thesis: E-bound P = 1
then A2: P c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by Th71;
A3: |[1,0]| in P by A1;
reconsider P = P as non empty Subset of (TOP-REAL 2) by A1;
reconsider Z = (proj1 | P) .: the carrier of ((TOP-REAL 2) | P) as Subset of REAL ;
A4: P = the carrier of ((TOP-REAL 2) | P) by PRE_TOPC:8;
A5: for p being Real st p in Z holds
p <= 1

proof

let p be Real; :: thesis: ( p in Z implies p <= 1 )
assume p in Z ; :: thesis: p <= 1
then consider p0 being object such that
A6: p0 in the carrier of ((TOP-REAL 2) | P) and
p0 in the carrier of ((TOP-REAL 2) | P) and
A7: p = (proj1 | P) . p0 by FUNCT_2:64;
p0 in closed_inside_of_rectangle ((- 1),1,(- 3),3) by A2, A4, A6;
then ex p1 being Point of (TOP-REAL 2) st
( p0 = p1 & - 1 <= p1 `1 & p1 `1 <= 1 & - 3 <= p1 `2 & p1 `2 <= 3 ) ;
hence p <= 1 by A4, A6, A7, PSCOMP_1:22; :: thesis: verum

end;

for q being Real st ( for p being Real st p in Z holds
p <= q ) holds
1 <= q

proof

let q be Real; :: thesis: ( ( for p being Real st p in Z holds
p <= q ) implies 1 <= q )
assume A8: for p being Real st p in Z holds
p <= q ; :: thesis: 1 <= q
(proj1 | P) . |[1,0]| = |[1,0]| `1 by A3, PSCOMP_1:22;
hence 1 <= q by A3, A4, A8, Lm17, FUNCT_2:35; :: thesis: verum

end;

hence E-bound P = 1 by A5, SEQ_4:46; :: thesis: verum