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;

theorem Th50:
  for p being Point of TOP-REAL 2 st p = CircleMap.r2 holds
  RotateCircle(1,-Arg(p)).(CircleMap.r1) = CircleMap.(r1-r2)
  proof
    let p be Point of T2;
    assume
A1: p = CM.r2;
    set s = -Arg(p);
    reconsider q = CM.R^1(r1), w = CM.R^1(r1-r2) as Point of T2
    by PRE_TOPC:25;
    |.q.| = 1 by TOPREALB:12;
    then q <> 0.T2 by TOPRNS_1:23;
    then consider i such that
A2: Arg((Rotate(s)).q) = s+(Arg q)+2*PI*i by Th43;
A3: Arg(p) = 2*PI*frac(r2) by A1,Th36;
A4: |.(Rotate(s)).q.| = |.q.| by Th41
    .= 1 by TOPREALB:12
    .= |.w.| by TOPREALB:12;
    consider j such that
A5: frac(r1-r2) = frac(r1) - frac(r2) + j and
    j = 0 or j = 1 by Th4;
A6: Arg (Rotate(s)).q = -(2*PI*frac(r2))+2*PI*frac(r1)+2*PI*i by A2,A3,Th36
    .= 2*PI*frac(r1-r2) + 2*PI*(i-j) by A5
    .= Arg w + 2*PI*(i-j) by Th36;
    thus RotateCircle(1,s).(CM.r1) = (Rotate(s)).q by FUNCT_1:49
    .= CM.(r1-r2) by A4,A6,Th35;
  end;
