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 T111:
  for f1,f2 being complex-valued Function holds
  r(#)(f1 + f2) = r(#)f1 + r(#)f2
proof
  let f1,f2 be complex-valued Function;
  thus
A1: dom (r(#)(f1 + f2)) = dom (f1 + f2) by VALUED_1:def 5
    .= dom f1 /\ dom f2 by VALUED_1:def 1
    .= dom f1 /\ dom (r(#)f2) by VALUED_1:def 5
    .= dom (r(#)f1) /\ dom (r(#)f2) by VALUED_1:def 5
    .= dom (r(#)f1 + r(#)f2) by VALUED_1:def 1;
    let c be object;
    assume
A2: c in dom (r(#)(f1 + f2));
    then
A3: c in dom (f1 + f2) by VALUED_1:def 5;
A4: c in dom (r(#)f1) /\ dom (r(#)f2) by A1,A2,VALUED_1:def 1;
    then
A5: c in dom (r(#)f1) by XBOOLE_0:def 4;
A6: c in dom (r(#)f2) by A4,XBOOLE_0:def 4;
    thus (r(#)(f1 + f2)).c = r * ((f1 + f2).c) by A2,VALUED_1:def 5
      .= r * (((f1.c)) + ((f2.c))) by A3,VALUED_1:def 1
      .= r * ((f1.c)) + r * ((f2.c))
      .= (r(#)f1).c + r * ((f2.c)) by A5,VALUED_1:def 5
      .= (r(#)f1).c + (r(#)f2).c by A6,VALUED_1:def 5
      .= (r(#)f1 + r(#)f2).c by A1,A2,VALUED_1:def 1;
end;
