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 Th46:
  for s being Real, p being Point of TOP-REAL 2 holds
  (Rotate(s)).((Rotate(-s)).p) = p
  proof
    let s be Real;
    let p be Point of T2;
    set f = Rotate(s);
    set g = Rotate(-s);
    per cases;
    suppose
A1:   p <> 0.T2;
      then consider i such that
A2:   Arg(g.p) = -s+(Arg p)+2*PI*i by Th43;
      consider j such that
A3:   Arg(f.(g.p)) = s+(Arg(g.p))+2*PI*j by A1,Th45,Th43;
A4:   |.f.(g.p).| = |.g.p.| by Th41
      .= |.p.| by Th41;
      Arg(f.(g.p)) = Arg p + 2*PI*(i+j) by A2,A3;
      hence f.(g.p) = p by A4,Th35;
    end;
    suppose
A5:   p = 0.T2;
      (Rotate(s)).((Rotate(-s)).(0.T2)) = (Rotate(s)).((0.T2)) by Th44
      .= 0.T2 by Th44;
      hence thesis by A5;
    end;
  end;
