reserve p1, p2 for Point of TOP-REAL 2,
  C for Simple_closed_curve,
  P for Subset of TOP-REAL 2;

theorem Th1:
  for C being non empty compact Subset of TOP-REAL 2 ex p1,p2 st p1
  ,p2 realize-max-dist-in C
proof
  let C be non empty compact Subset of TOP-REAL 2;
  reconsider D=C as Subset of TopSpaceMetr Euclid 2 by Lm1;
A1: D is compact by Lm1,COMPTS_1:23;
  then consider x1,x2 being Point of Euclid 2 such that
A2: x1 in D & x2 in D and
A3: dist(x1,x2) = max_dist_max(D,D) by WEIERSTR:33;
  reconsider a=x1,b=x2 as Point of TOP-REAL 2 by EUCLID:67;
  take a,b;
  thus a in C & b in C by A2;
  let x, y be Point of TOP-REAL 2 such that
A4: x in C & y in C;
  reconsider x9=x,y9=y as Point of Euclid 2 by EUCLID:67;
  dist(x9,y9) <= max_dist_max(D,D) by A1,A4,WEIERSTR:34;
  then dist(x,y) <= max_dist_max(D,D) by TOPREAL6:def 1;
  hence thesis by A3,TOPREAL6:def 1;
end;
