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,V
  for f3 be PartFunc of C,REAL holds
  f3(#)f1 - f3(#)f2 = f3(#)(f1 - f2)
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,V;
  let f3 be PartFunc of C,REAL;
A1: dom (f3 (#) f1 - f3 (#) f2) = dom (f3 (#) f1) /\ dom (f3 (#) f2) by Def2
    .= dom (f3 (#) f1) /\ (dom f3 /\ dom f2) by Def3
    .= dom f3 /\ dom f1 /\ (dom f3 /\ dom f2) by Def3
    .= dom f3 /\ (dom f3 /\ dom f1) /\ dom f2 by XBOOLE_1:16
    .= dom f3 /\ dom f3 /\ dom f1 /\ dom f2 by XBOOLE_1:16
    .= dom f3 /\ (dom f1 /\ dom f2) by XBOOLE_1:16
    .= dom f3 /\ dom (f1 - f2) by Def2
    .= dom (f3(#)(f1 - f2)) by Def3;
  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,Def2;
    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,Def2
      .= f3.c * (f1/.c) - f3.c * (f2/.c) by RLVECT_1:34
      .= ((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,Def2;
  end;
  hence thesis by A1,PARTFUN2:1;
end;
