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 scalar-associative non empty RLSStruct
  for f1 being PartFunc of C,REAL
  for f2 being PartFunc of C,V holds
  r(#)(f1(#)f2) = r(#)f1(#)f2
proof
  let V be scalar-associative non empty RLSStruct;
  let f1 be PartFunc of C,REAL;
  let f2 be PartFunc of C,V;
A1: dom (r(#)(f1 (#) f2)) = dom (f1 (#) f2) by Def4
    .= dom f1 /\ dom f2 by Def3
    .= dom (r(#)f1) /\ dom f2 by VALUED_1:def 5
    .= dom (r(#)f1(#)f2) by Def3;
  now
    let c;
    assume
A2: c in dom (r(#)(f1(#)f2));
    then
A3: c in dom (f1(#)f2) by Def4;
    c in dom (r(#)f1) /\ dom f2 by A1,A2,Def3;
    then
A4: c in dom (r(#)f1) by XBOOLE_0:def 4;
    thus (r(#)(f1(#)f2))/.c = r * ((f1(#)f2)/.c) by A2,Def4
      .= r*(f1.c * (f2/.c)) by A3,Def3
      .= (r*f1.c) * (f2/.c) by RLVECT_1:def 7
      .= (r(#)f1).c * (f2/.c) by A4,VALUED_1:def 5
      .= (r(#)f1 (#) f2)/.c by A1,A2,Def3;
  end;
  hence thesis by A1,PARTFUN2:1;
end;
