reserve B for non empty set,
  A,X,x for set;
reserve phi for Element of Funcs(A,B);

theorem Th4:
  for X,Y being set st Funcs(X,Y) <> {} & X c= A & Y c= B for f
  being Element of Funcs(X,Y) ex phi being Element of Funcs(A,B) st phi|X = f
proof
  let X,Y be set such that
A1: Funcs(X,Y) <> {} and
A2: X c= A and
A3: Y c= B;
  let f be Element of Funcs(X,Y);
  reconsider f9=f as PartFunc of A,B by A1,A2,A3,Th3;
  consider phi being Function of A,B such that
A4: for x being object st x in dom f9 holds phi.x = f9.x by FUNCT_2:71;
  reconsider phi as Element of Funcs(A,B) by FUNCT_2:8;
  take phi;
  ex g being Function st f = g & dom g = X & rng g c= Y by A1,FUNCT_2:def 2;
  then dom f9 = A /\ X by XBOOLE_1:28
    .= dom phi /\ X by FUNCT_2:def 1;
  hence thesis by A4,FUNCT_1:46;
end;
