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 add-associative right_zeroed right_complementable Abelian
    scalar-distributive scalar-associative scalar-unital vector-distributive
    non empty RLSStruct
  for f1,f2 being PartFunc of C,REAL
  for f3 being Function of C,V holds
  (f1 - f2)(#)f3 = f1(#)f3 - f2(#)f3
proof
  let V be add-associative right_zeroed right_complementable Abelian
  scalar-distributive scalar-associative scalar-unital vector-distributive
  non empty RLSStruct;
  let f1,f2 be PartFunc of C,REAL;
  let f3 be Function of C,V;
A1: dom ((f1 - f2) (#) f3) = dom (f1 - f2) /\ dom f3 by Def3
    .= dom f1 /\ dom f2 /\ (dom f3 /\ dom f3) by VALUED_1:12
    .= dom f1 /\ dom f2 /\ dom f3 /\ dom f3 by XBOOLE_1:16
    .= dom f1 /\ dom f3 /\ dom f2 /\ dom f3 by XBOOLE_1:16
    .= dom f1 /\ dom f3 /\ (dom f2 /\ dom f3) by XBOOLE_1:16
    .= dom (f1 (#) f3) /\ (dom f2 /\ dom f3) by Def3
    .= dom (f1 (#) f3) /\ dom (f2 (#) f3) by Def3
    .= dom (f1 (#) f3 - f2 (#) f3) by Def2;
  now
    let c;
    assume
A2: c in dom ((f1 - f2)(#)f3);
    then c in dom (f1 - f2) /\ dom f3 by Def3;
    then
A3: c in dom (f1 - f2) by XBOOLE_0:def 4;
A4: c in dom (f1(#)f3) /\ dom (f2(#)f3) by A1,A2,Def2;
    then
A5: c in dom (f1(#)f3) by XBOOLE_0:def 4;
A6: c in dom (f2 (#) f3) by A4,XBOOLE_0:def 4;
    thus ((f1 - f2) (#) f3)/.c = (f1 - f2).c * (f3/.c) by A2,Def3
      .= (f1.c - f2.c) * (f3/.c) by A3,VALUED_1:13
      .= f1.c * (f3/.c) - f2.c * (f3/.c) by RLVECT_1:35
      .= ((f1 (#) f3)/.c) - f2.c * (f3/.c) by A5,Def3
      .= ((f1 (#) f3)/.c) - ((f2 (#) f3)/.c) by A6,Def3
      .= ((f1 (#) f3) - (f2 (#) f3))/.c by A1,A2,Def2;
  end;
  hence thesis by A1,PARTFUN2:1;
end;
