reserve n for Element of NAT,
  i for Integer,
  a, b, r for Real,
  x for Point of TOP-REAL n;

theorem
  for i, j being Integer holds CircleMap.r = |[ (cos*AffineMap(2*PI,2*PI
  *i)).r, (sin*AffineMap(2*PI,2*PI*j)).r ]|
proof
  let i, j be Integer;
  thus CircleMap.r = |[ cos(2*PI*r+0), sin(2*PI*r) ]| by Def11
    .= |[ cos(2*PI*r+2*PI*i), sin(2*PI*r+0) ]| by COMPLEX2:9
    .= |[ cos(2*PI*r+2*PI*i), sin(2*PI*r+2*PI*j) ]| by COMPLEX2:8
    .= |[ (cos*AffineMap(2*PI,2*PI*i)).r, sin(2*PI*r+2*PI*j) ]| by Th2
    .= |[ (cos*AffineMap(2*PI,2*PI*i)).r, (sin*AffineMap(2*PI,2*PI*j)).r ]|
  by Th1;
end;
