reserve x for object, X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for complex-valued Function;
reserve r,p for Complex;

theorem Th28:
  r<>0 implies (r(#)f)^ = r" (#) (f^)
proof
  assume
A1: r<>0;
A2: dom ((r(#)f)^) = dom (r(#)f) \ (r(#)f)"{0} by Def2
    .= dom (r(#)f) \ f"{0} by A1,Th7
    .= dom f \ f"{0} by VALUED_1:def 5
    .= dom (f^) by Def2
    .= dom (r"(#)(f^)) by VALUED_1:def 5;
  now
    let c be object;
    assume
A3: c in dom ((r(#)f)^);
    then
A4: c in dom (r(#)f) \ (r(#)f)"{0} by Def2;
A5: c in dom (f^) by A2,A3,VALUED_1:def 5;
    thus ((r(#)f)^).c = ((r(#)f).c)" by A3,Def2
      .= (r*(f.c))" by A4,VALUED_1:def 5
      .= r"* (f.c)" by XCMPLX_1:204
      .= r"* ((f^).c) by A5,Def2
      .= (r" (#) (f^)).c by A2,A3,VALUED_1:def 5;
  end;
  hence thesis by A2,FUNCT_1:2;
end;
