reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;
reserve D for non empty set;
reserve A,B for non empty set;

theorem
  for f being Function of A,B, A0 being non empty Subset of A, f0 being
Function of A0,B st for c being Element of A st c in A0 holds f.c = f0.c holds
  f|A0 = f0
proof
  let f be Function of A,B, A0 be non empty Subset of A, f0 be Function of A0,
  B such that
A1: for c being Element of A st c in A0 holds f.c = f0.c;
  reconsider g = f|A0 as Function of A0,B by Th32;
  for c being Element of A0 holds g.c = f0.c
  proof
    let c be Element of A0;
    thus g.c = f.c by FUNCT_1:49
      .= f0.c by A1;
  end;
  hence thesis by Th62;
end;
