theorem Th57:
  for f being complex-valued Function holds f" ^2 = f^2 "
  proof
    let f be complex-valued Function;
A1: dom(f" ^2) = dom(f") by VALUED_1:11;
A2: dom(f") = dom(f) by VALUED_1:def 7;
    dom(f^2 ") = dom(f^2) by VALUED_1:def 7;
    hence dom(f" ^2) = dom(f^2 ") by A1,A2,VALUED_1:11;
    let x;
    assume x in dom(f" ^2);
A3: (f^2).x = (f.x)^2 by VALUED_1:11;
    f".x = (f.x)" by VALUED_1:10;
    hence (f" ^2).x = (1/(f.x)) ^2 by VALUED_1:11
    .= (1^2)/((f.x)^2) by XCMPLX_1:76
    .= (f^2 ").x by A3,VALUED_1:10;
  end;
