 reserve a,Z1,Z2,Z3 for set,
         x,y,z for object,
         k for Nat;
 reserve S for RelStr;
 reserve P,Q for non empty flat Poset;
 reserve p,p1,p2 for Element of P;
 reserve K for non empty Chain of P;

theorem Thflat0501:
  for f being Function of P,Q holds
    f.(Bottom P) = Bottom Q implies f is monotone
  proof
    let f be Function of P,Q;
    assume
A1: f.(Bottom P) = Bottom Q;
    set z = Bottom P;
    for p1,p2 being Element of P st p1 <= p2
    for q1,q2 being Element of Q st q1 = f.p1 & q2 = f.p2
         holds q1 <= q2
    proof
      let p1,p2 be Element of P;
      assume p1 <= p2;
      then p1 = z or p1 = p2 by Lemflat01;
      hence thesis by A1,Lemflat01;
    end;
    hence thesis;
  end;
