reserve a,b,c,x,y,z for object,X,Y,Z for set,
  n for Nat,
  i,j for Integer,
  r,r1,r2,r3,s for Real,
  c1,c2 for Complex,
  p for Point of TOP-REAL n;
reserve n for non zero Nat;

theorem Th56:
  for f being Function of Tcircle(0.TOP-REAL(n+1),1),TOP-REAL n
  for x being Point of Tcircle(0.TOP-REAL(n+1),1) st
  f is without_antipodals holds f.x - f.-x <> 0.TOP-REAL n
  proof
    set TC4 = Tcircle(0.TOP-REAL(n+1),1);
    let f be Function of TC4,TOP-REAL n;
    let x be Point of TC4;
    assume
A1: f is without_antipodals;
A2: dom f = the carrier of TC4 by FUNCT_2:def 1;
    reconsider y = -x as Point of TC4 by TOPREALC:60;
    reconsider a = x, b = y as Point of TC4;
    reconsider x1 = x as Point of TOP-REAL(n+1) by PRE_TOPC:25;
    assume
A3: f.x-f.-x = 0.TOP-REAL n;
    x1,-x1 are_antipodals_of 0.TOP-REAL(n+1),1,f
    proof
      thus x1,-x1 are_antipodals_of 0.TOP-REAL(n+1),1 by Th54;
      f.x = f.a & f.y = f.b;
      hence x1 in dom f & -x1 in dom f by A2;
      f.a-f.y = 0.TOP-REAL n by A3;
      hence f.x1 = f.-x1 by RLVECT_1:21;
    end;
    hence contradiction by A1;
  end;
