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,f2 being PartFunc of C,REAL, f3 being PartFunc of C,V
  holds (f1 (#) f2) (#) f3 = f1 (#) (f2 (#) f3)
proof
  let V be scalar-associative non empty RLSStruct;
  let f1,f2 be PartFunc of C,REAL;
  let f3 be PartFunc of C,V;
A1: dom (f1 (#) f2 (#) f3) = dom (f1 (#) f2) /\ dom f3 by Def3
    .= dom f1 /\ dom f2 /\ dom f3 by VALUED_1:def 4
    .= dom f1 /\ (dom f2 /\ dom f3) by XBOOLE_1:16
    .= dom f1 /\ dom (f2 (#) f3) by Def3
    .= dom (f1 (#) (f2 (#) f3)) by Def3;
  now
    let c;
    assume
A2: c in dom (f1(#)f2(#)f3);
    then c in dom f1 /\ dom (f2(#)f3) by A1,Def3;
    then
A3: c in dom (f2 (#) f3) by XBOOLE_0:def 4;
    thus (f1 (#) f2 (#) f3)/.c = (f1 (#) f2).c * (f3/.c) by A2,Def3
      .= f1.c * f2.c * (f3/.c) by VALUED_1:5
      .= f1.c * (f2.c * (f3/.c)) by RLVECT_1:def 7
      .= f1.c * ((f2 (#) f3)/.c) by A3,Def3
      .= (f1 (#) (f2 (#) f3))/.c by A1,A2,Def3;
  end;
  hence thesis by A1,PARTFUN2:1;
end;
