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 Th49:
  for r being non negative Real, s being Real holds
  (Rotate(s)).:Sphere(0.TOP-REAL 2,r) = Sphere(0.TOP-REAL 2,r)
  proof
    let r be non negative Real;
    let s be Real;
    set f = Rotate(s);
    set C = Sphere(0.T2,r);
    thus f.:C c= C
    proof
      let y be Point of T2;
      assume y in f.:C;
      then ex c being Element of T2 st c in C & y = f.c by FUNCT_2:65;
      hence y in C by Th48;
    end;
    let y be Point of T2;
    set x = (Rotate(-s)).y;
    assume y in C;
    then x in C by Th48;
    then f.x in f.:C by FUNCT_2:35;
    hence y in f.:C by Th46;
  end;
