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 Th49:
  (r(#)f)|X = r(#)(f|X)
proof
A1: now
    let c be object;
    assume
A2: c in dom ((r(#)f)|X);
    then
A3: c in dom (r(#)f) /\ X by RELAT_1:61;
    then
A4: c in X by XBOOLE_0:def 4;
A5: c in dom (r(#)f) by A3,XBOOLE_0:def 4;
    then c in dom f by VALUED_1:def 5;
    then c in dom f /\ X by A4,XBOOLE_0:def 4;
    then
A6: c in dom (f|X) by RELAT_1:61;
    then
A7: c in dom (r(#)(f|X)) by VALUED_1:def 5;
    thus ((r(#)f)|X).c = (r(#)f).c by A2,FUNCT_1:47
      .= r*(f.c) by A5,VALUED_1:def 5
      .= r*(f|X).c by A6,FUNCT_1:47
      .= (r(#)(f|X)).c by A7,VALUED_1:def 5;
  end;
  dom ((r(#)f)|X) = dom (r(#)f) /\ X by RELAT_1:61
    .= dom f /\ X by VALUED_1:def 5
    .= dom (f|X) by RELAT_1:61
    .= dom (r(#)(f|X)) by VALUED_1:def 5;
  hence thesis by A1,FUNCT_1:2;
end;
