theorem Th48:
  for s being Real, p being Point of TOP-REAL 2 holds
  p in Sphere(0.TOP-REAL 2,r) iff (Rotate(s)).p in Sphere(0.TOP-REAL 2,r)
  proof
    let s be Real;
    let p be Point of T2;
A1: |.p.| = |.(Rotate(s)).p.| by Th41;
A2: (Rotate(s)).p-0.T2 = (Rotate(s)).p by RLVECT_1:13;
    hereby
      assume p in Sphere(0.T2,r);
      then |.p.| = r by TOPREAL9:12;
      hence (Rotate(s)).p in Sphere(0.T2,r) by A1,A2,TOPREAL9:9;
    end;
    assume (Rotate(s)).p in Sphere(0.T2,r);
    then
A3: |.(Rotate(s)).p.| = r by TOPREAL9:12;
A4: (Rotate(-s)).((Rotate(--s)).p) = p by Th46;
    (Rotate(-s)).((Rotate(s)).p)-0.T2 = (Rotate(-s)).((Rotate(s)).p)
    by RLVECT_1:13;
    hence p in Sphere(0.T2,r) by A4,A3,A1,TOPREAL9:9;
  end;
