theorem Th54:
  for n being non zero Nat,
      r being non negative Real,
      x being Point of TOP-REAL n st x is Point of Tcircle(0.TOP-REAL n,r)
  holds x,-x are_antipodals_of 0.TOP-REAL n,r
  proof
    let n be non zero Nat,
        r be non negative Real,
        x be Point of TOP-REAL n such that
A1:  x is Point of Tcircle(0.TOP-REAL n,r);
     reconsider y = x as Point of Tcircle(0.TOP-REAL n,r) by A1;
     -x = -y;
     hence x is Point of Tcircle(0.TOP-REAL n,r) &
     -x is Point of Tcircle(0.TOP-REAL n,r) by TOPREALC:60;
     (1-1/2)*x + (1/2)*(-x) = (1/2)*x +- (1/2)*x by RLVECT_1:25
     .= 0.TOP-REAL n by RLVECT_1:5;
     hence thesis;
   end;
