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 holds
  r(#)(f1 - f2) = r(#)f1 - r(#)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;
A1: dom (r(#)(f1 - f2)) = dom (f1 - f2) by Def4
    .= dom f1 /\ dom f2 by Def2
    .= dom f1 /\ dom (r(#)f2) by Def4
    .= dom (r(#)f1) /\ dom (r(#)f2) by Def4
    .= dom (r(#)f1 - r(#)f2) by Def2;
  now
    let c;
    assume
A2: c in dom (r(#)(f1 - f2));
    then
A3: c in dom (f1 - f2) by Def4;
A4: c in dom (r(#)f1) /\ dom (r(#)f2) by A1,A2,Def2;
    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,Def4
      .= r * ((f1/.c) - (f2/.c)) by A3,Def2
      .= r * (f1/.c) - r * (f2/.c) by RLVECT_1:34
      .= ((r(#)f1)/.c) - r * (f2/.c) by A5,Def4
      .= ((r(#)f1)/.c) - ((r(#)f2)/.c) by A6,Def4
      .= (r(#)f1 - r(#)f2)/.c by A1,A2,Def2;
  end;
  hence thesis by A1,PARTFUN2:1;
end;
