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 Th55:
  for n being non zero Nat,
      p, x, y, x1, y1 being Point of TOP-REAL n,
      r being positive Real
  st x,y are_antipodals_of 0.TOP-REAL n,1 &
     x1 = CircleIso(p,r).x & y1 = CircleIso(p,r).y
  holds x1,y1 are_antipodals_of p,r
  proof
    let n be non zero Nat,
        p, x, y, x1, y1 be Point of TOP-REAL n,
        r be positive Real;
    set h = CircleIso(p,r);
    assume that
A1: x,y are_antipodals_of 0.TOP-REAL n,1 and
A2: x1 = h.x and
A3: y1 = h.y;
A4: x is Point of Tcircle(0.TOP-REAL n,1) by A1;
    hence x1 is Point of Tcircle(p,r) by A2,FUNCT_2:5;
A5: y is Point of Tcircle(0.TOP-REAL n,1) by A1;
    hence y1 is Point of Tcircle(p,r) by A3,FUNCT_2:5;
    0.TOP-REAL n in LSeg(x,y) by A1;
    then consider a being Real such that
A6: 0.TOP-REAL n = (1-a)*x + a*y and
A7: 0 <= a & a <= 1;
A8: (1-a)*x1 = (1-a)*(r*x+p) by A2,A4,Def5
    .= (1-a)*(r*x)+(1-a)*p by RLVECT_1:def 5
    .= r*(1-a)*x+(1-a)*p by RLVECT_1:def 7
    .= r*((1-a)*x)+(1-a)*p by RLVECT_1:def 7;
    a*y1 = a*(r*y+p) by A3,A5,Def5
    .= a*(r*y)+a*p by RLVECT_1:def 5
    .= r*a*y+a*p by RLVECT_1:def 7
    .= r*(a*y)+a*p by RLVECT_1:def 7;
    then (1-a)*x1 + a*y1 = r*((1-a)*x)+(1-a)*p + r*(a*y)+a*p
    by A8,RLVECT_1:def 3
    .= r*((1-a)*x) + r*(a*y) + (1-a)*p + a*p by RLVECT_1:def 3
    .= r*((1-a)*x+(a*y)) + (1-a)*p + a*p by RLVECT_1:def 5
    .= 0.TOP-REAL n + (1-a)*p + a*p by A6,RLVECT_1:10
    .= (1-a)*p + a*p by RLVECT_1:4
    .= 1*p - a*p + a*p by RLVECT_1:35
    .= 1*p by RLVECT_4:1
    .= p by RLVECT_1:def 8;
    hence thesis by A7;
  end;
