reserve a, b, c, d, r, s for Real,
  n for Element of NAT,
  p, p1, p2 for Point of TOP-REAL 2,
  x, y for Point of TOP-REAL n,
  C for Simple_closed_curve,
  A, B, P for Subset of TOP-REAL 2,
  U, V for Subset of (TOP-REAL 2)|C`,
  D for compact with_the_max_arc Subset of TOP-REAL 2;

theorem Th78:
  for P being compact Subset of TOP-REAL 2 holds
  |[-1,0]|,|[1,0]| realize-max-dist-in P implies E-most P = {|[1,0]|}
proof
  let P be compact Subset of T2;
  assume
A1: a,b realize-max-dist-in P;
  then
A2: P c= R by Th71;
  set L = LSeg(SE-corner P, NE-corner P);
A3: b in P by A1;
A4: (SE-corner P)`1 = |[rp,S-bound P]|`1 by A1,Th76
    .= rp by EUCLID:52;
A5: (NE-corner P)`1 = |[rp,N-bound P]|`1 by A1,Th76
    .= rp by EUCLID:52;
  thus E-most P c= {b}
  proof
    let x be object;
    assume
A6: x in E-most P;
    then
A7: x in P by XBOOLE_0:def 4;
    reconsider x as Point of T2 by A6;
A8: x in L by A6,XBOOLE_0:def 4;
    SE-corner P in L by RLTOPSP1:68;
    then
A9: x`1 = rp by A4,A8,SPPOL_1:def 3;
    x in R by A2,A7;
    then ex p st x = p & rl <= p`1 & p`1 <= rp & rd <= p`2 & p`2 <= rg;
    then x in dR by A9,Lm61;
    then x in P /\ dR by A7,XBOOLE_0:def 4;
    then x in {a,b} by A1,Th74;
    then x = a or x = b by TARSKI:def 2;
    hence thesis by A9,EUCLID:52,TARSKI:def 1;
  end;
  let x be object;
  assume x in {b};
  then
A10: x = b by TARSKI:def 1;
A11: (SE-corner P)`2 = S-bound P by EUCLID:52;
A12: (NE-corner P)`2 = N-bound P by EUCLID:52;
A13: (SE-corner P)`2 <= b`2 by A3,A11,PSCOMP_1:24;
  b`2 <= (NE-corner P)`2 by A3,A12,PSCOMP_1:24;
  then b in L by A4,A5,A13,Lm17,GOBOARD7:7;
  hence thesis by A3,A10,XBOOLE_0:def 4;
end;
