reserve x,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve V for RealNormSpace;
reserve f,f1,f2,f3 for PartFunc of C,V;
reserve r,r1,r2,p for Real;

theorem
  for V being vector-distributive non empty RLSStruct
  for f1,f2 being PartFunc of C,V
  for f3 being Function of C,REAL holds
  f3 (#) (f1 + f2) = f3(#)f1 + f3(#)f2
proof
  let V be vector-distributive non empty RLSStruct;
  let f1,f2 be PartFunc of C,V;
  let f3 be Function of C,REAL;
A1: dom (f3 (#) (f1 + f2)) = dom f3 /\ dom (f1 + f2) by Def3
    .= dom f3 /\ (dom f1 /\ dom f2) by Def1
    .= dom f3 /\ dom f3 /\ dom f1 /\ dom f2 by XBOOLE_1:16
    .= dom f3 /\ dom f1 /\ dom f3 /\ dom f2 by XBOOLE_1:16
    .= dom f3 /\ dom f1 /\ (dom f3 /\ dom f2) by XBOOLE_1:16
    .= dom (f3 (#) f1) /\ (dom f3 /\ dom f2) by Def3
    .= dom (f3 (#) f1) /\ dom (f3 (#) f2) by Def3
    .= dom (f3 (#) f1 + f3 (#) f2) by Def1;
  now
    let c;
    assume
A2: c in dom (f3(#)(f1 + f2));
    then c in dom f3 /\ dom (f1 + f2) by Def3;
    then
A3: c in dom (f1 + f2) by XBOOLE_0:def 4;
A4: c in dom (f3(#)f1) /\ dom (f3(#)f2) by A1,A2,Def1;
    then
A5: c in dom (f3(#)f1) by XBOOLE_0:def 4;
A6: c in dom (f3 (#) f2) by A4,XBOOLE_0:def 4;
    thus (f3 (#) (f1 + f2))/.c = f3.c * ((f1 + f2)/.c) by A2,Def3
      .= f3.c * ((f1/.c) + (f2/.c)) by A3,Def1
      .= f3.c * (f1/.c) + f3.c * (f2/.c) by RLVECT_1:def 5
      .= ((f3 (#) f1)/.c) + f3.c* (f2/.c) by A5,Def3
      .= ((f3 (#) f1)/.c) + ((f3 (#) f2)/.c) by A6,Def3
      .= ((f3 (#) f1) + (f3 (#) f2))/.c by A1,A2,Def1;
  end;
  hence thesis by A1,PARTFUN2:1;
end;
