reserve C for Simple_closed_curve,
  i, j, n for Nat,
  p for Point of TOP-REAL 2;

theorem
  for C being compact Subset of TOP-REAL 2 holds p in BDD C implies p`1
  <> E-bound BDD C
proof
  reconsider P = p as Point of Euclid 2 by Lm3;
  let C be compact Subset of TOP-REAL 2;
A1: BDD C is bounded by JORDAN2C:106;
  assume p in BDD C;
  then consider r being Real such that
A2: r > 0 and
A3: Ball(P,r) c= BDD C by Th17;
  set EX = |[p`1 + r/2, p`2]|;
  0 < r/2 by A2,XREAL_1:139;
  then
A4: p`1 + r/2 > p`1 + 0 by XREAL_1:6;
  assume
A5: p`1 = E-bound BDD C;
A6: not EX in BDD C
  proof
    assume EX in BDD C;
    then EX`1 <= E-bound BDD C by A1,Th5;
    hence thesis by A5,A4,EUCLID:52;
  end;
A7: P = |[p`1,p`2]| by EUCLID:53;
  r/2 < r by A2,XREAL_1:216;
  then EX in Ball(P,r) by A2,A7,GOBOARD6:7;
  hence thesis by A3,A6;
end;
