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 Th52:
  f|Y is constant implies (z(#)f)|Y is constant
proof
  assume f|Y is constant;
  then consider r being VECTOR of V such that
A1: for c be Element of M st c in Y /\ dom f holds f/.c = r by PARTFUN2:35;
  now
    let c be Element of M;
    assume
A2: c in Y /\ dom (z(#)f);
    then
A3: c in Y by XBOOLE_0:def 4;
A4: c in dom (z(#)f) by A2,XBOOLE_0:def 4;
    then c in dom f by Def2;
    then
A5: c in Y /\ dom f by A3,XBOOLE_0:def 4;
    thus (z(#)f)/.c = z * f/.c by A4,Def2
      .= z*r by A1,A5;
  end;
  hence thesis by PARTFUN2:35;
end;
