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
  (z(#)f)|X = z(#)(f|X)
proof
A1: now
    let c be Element of M;
    assume
A2: c in dom ((z(#)f)|X);
    then
A3: c in dom (z(#)f) /\ X by RELAT_1:61;
    then
A4: c in X by XBOOLE_0:def 4;
A5: c in dom (z(#)f) by A3,XBOOLE_0:def 4;
    then c in dom f by Def2;
    then c in dom f /\ X by A4,XBOOLE_0:def 4;
    then
A6: c in dom (f|X) by RELAT_1:61;
    then
A7: c in dom (z(#)(f|X)) by Def2;
    thus ((z(#)f)|X)/.c = (z(#)f)/.c by A2,PARTFUN2:15
      .= z*(f/.c) by A5,Def2
      .= z*((f|X)/.c) by A6,PARTFUN2:15
      .= (z(#)(f|X))/.c by A7,Def2;
  end;
  dom ((z(#)f)|X) = dom (z(#)f) /\ X by RELAT_1:61
    .= dom f /\ X by Def2
    .= dom (f|X) by RELAT_1:61
    .= dom (z(#)(f|X)) by Def2;
  hence thesis by A1,PARTFUN2:1;
end;
