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 Th34:
  (f1(#)f2)^ = (f1^)(#)(f2^)
proof
A1: dom ((f1(#)f2)^) = dom (f1(#)f2) \ (f1(#)f2)"{0c} by Def2
    .= (dom f1 \ f1"{0c}) /\ (dom f2 \ (f2)"{0c}) by Th7
    .= dom (f1^) /\ (dom f2 \ (f2)"{0c}) by Def2
    .= dom (f1^) /\ dom (f2^) by Def2
    .= dom ((f1^) (#) (f2^)) by Th3;
  now
    let c;
    assume
A2: c in dom ((f1(#)f2)^);
    then c in dom (f1(#)f2) \ (f1(#)f2)"{0c} by Def2;
    then
A3: c in dom (f1(#)f2) by XBOOLE_0:def 5;
A4: c in dom (f1^) /\ dom (f2^) by A1,A2,Th3;
    then
A5: c in dom (f1^) by XBOOLE_0:def 4;
A6: c in dom (f2^) by A4,XBOOLE_0:def 4;
    thus ((f1(#)f2)^)/.c = ((f1(#)f2)/.c)" by A2,Def2
      .= (((f1/.c)) * ((f2/.c)))" by A3,Th3
      .= (((f1/.c)))"* (((f2/.c)))" by XCMPLX_1:204
      .= ((f1^)/.c)*(((f2/.c)))" by A5,Def2
      .= ((f1^)/.c) *((f2^)/.c) by A6,Def2
      .= ((f1^) (#) (f2^))/.c by A1,A2,Th3;
  end;
  hence thesis by A1,PARTFUN2:1;
end;
