reserve x,y,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for PartFunc of C,COMPLEX;
reserve r1,r2,p1 for Real;
reserve r,q,cr1,cr2 for Complex;

theorem Th35:
  r <> 0 implies (r(#)f)^ = r" (#) (f^)
proof
  assume
A1: r<>0;
A2: dom ((r(#)f)^) = dom (r(#)f) \ (r(#)f)"{0c} by Def2
    .= dom (r(#)f) \ f"{0c} by A1,Th12
    .= dom f \ f"{0c} by Th4
    .= dom (f^) by Def2
    .= dom (r"(#)(f^)) by Th4;
  now
    let c;
    assume
A3: c in dom ((r(#)f)^);
    then
A4: c in dom (f^) by A2,Th4;
    c in dom (r(#)f) \ (r(#)f)"{0c} by A3,Def2;
    then
A5: c in dom (r(#)f) by XBOOLE_0:def 5;
    thus ((r(#)f)^)/.c = ((r(#)f)/.c)" by A3,Def2
      .= (r*((f/.c)))" by A5,Th4
      .= r"* ((f/.c))" by XCMPLX_1:204
      .= r"* ((f^)/.c) by A4,Def2
      .= (r" (#) (f^))/.c by A2,A3,Th4;
  end;
  hence thesis by A2,PARTFUN2:1;
end;
