reserve x for object, X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for complex-valued Function;
reserve r,p for Complex;

theorem Th9:
  (f1 (#) f2) (#) f3 = f1 (#) (f2 (#) f3)
proof
A1: now
    let c be object;
    assume c in dom (f1(#)f2(#)f3);
    thus (f1 (#) f2 (#) f3).c = (f1 (#) f2).c * f3.c by VALUED_1:5
      .= f1.c * f2.c * f3.c by VALUED_1:5
      .= f1.c * (f2.c * f3.c)
      .= f1.c * (f2 (#) f3).c by VALUED_1:5
      .= (f1 (#) (f2 (#) f3)).c by VALUED_1:5;
  end;
  dom (f1 (#) f2 (#) f3) = dom (f1 (#) f2) /\ dom f3 by VALUED_1:def 4
    .= 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 VALUED_1:def 4
    .= dom (f1 (#) (f2 (#) f3)) by VALUED_1:def 4;
  hence thesis by A1,FUNCT_1:2;
end;
