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 f being PartFunc of C,V holds
  (r*p)(#)f = r(#)(p(#)f)
proof
  let V be scalar-associative non empty RLSStruct;
  let f be PartFunc of C,V;
A1: dom ((r*p) (#) f) = dom f by Def4
    .= dom (p(#)f) by Def4
    .= dom (r(#)(p(#)f)) by Def4;
  now
    let c;
    assume
A2: c in dom ((r*p)(#)f);
    then
A3: c in dom (p(#)f) by A1,Def4;
    thus ((r*p)(#)f)/.c = r*p * f/.c by A2,Def4
      .= r*(p * f/.c) by RLVECT_1:def 7
      .= r * ((p(#)f)/.c) by A3,Def4
      .= (r(#)(p(#)f))/.c by A1,A2,Def4;
  end;
  hence thesis by A1,PARTFUN2:1;
end;
