reserve M for non empty set;
reserve V for ComplexNormSpace;
reserve f,f1,f2,f3 for PartFunc of M,V;
reserve z,z1,z2 for Complex;
reserve X,Y for set;

theorem
  for f1 be PartFunc of M,COMPLEX holds f1|X is constant & f2|Y is
  constant implies (f1(#)f2)|(X /\ Y) is constant
proof
  let f1 be PartFunc of M,COMPLEX;
  assume that
A1: f1|X is constant and
A2: f2|Y is constant;
  consider z1 be Element of COMPLEX such that
A3: for c be Element of M st c in X /\ dom f1 holds f1.c = z1
  by A1,PARTFUN2:57;
  consider r2 being VECTOR of V such that
A4: for c be Element of M st c in Y /\ dom f2 holds (f2/.c) = r2 by A2,
PARTFUN2:35;
  now
    let c be Element of M;
    assume
A5: c in X /\ Y /\ dom (f1(#)f2);
    then
A6: c in X /\ Y by XBOOLE_0:def 4;
    then
A7: c in Y by XBOOLE_0:def 4;
A8: c in dom (f1(#)f2) by A5,XBOOLE_0:def 4;
    then
A9: c in (dom f1 /\ dom f2) by Def1;
    then
A10: c in dom f1 by XBOOLE_0:def 4;
    then
A11: f1/.c = f1.c by PARTFUN1:def 6;
    c in dom f2 by A9,XBOOLE_0:def 4;
    then
A12: c in Y /\ dom f2 by A7,XBOOLE_0:def 4;
    c in X by A6,XBOOLE_0:def 4;
    then
A13: c in X /\ dom f1 by A10,XBOOLE_0:def 4;
    thus (f1(#)f2)/.c = f1/.c * (f2/.c) by A8,Def1
      .= z1 * (f2/.c) by A3,A13,A11
      .= z1 * r2 by A4,A12;
  end;
  hence thesis by PARTFUN2:35;
end;
