 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 Thflat07:
  for P,Q being strict non empty chain-complete flat Poset,
      f being Function of P,Q
     st f.(Bottom P) = Bottom Q holds f is continuous
  proof
    let P,Q be strict non empty chain-complete flat Poset;
    let f be Function of P,Q;
    assume
A1:   f.(Bottom P) = Bottom Q;
    then reconsider f as monotone Function of P,Q by Thflat0501;
    for K being non empty Chain of P holds f.(sup K) <= sup (f.:K)
    proof
      let K be non empty Chain of P;
      reconsider M = f.:K as non empty Chain of Q by POSET_1:1;
      consider a being Element of P such that
B1:     (K = {a} & f.:K = {f.a}) or
        (K = {Bottom P, a} & f.:K = {f.(Bottom P), f.a}) by Thflat05;
      per cases by A1,B1;
        suppose K = {a} & M = {f.a};
          then sup K =a & sup M = f.a by YELLOW_0:39;
          hence thesis;
          end;
        suppose K = {Bottom P, a} & M = {Bottom Q, f.a};
          then sup K = a & sup M = f.a by Thflat0502;
          hence thesis;
      end;
    end;
    hence thesis by POSET_1:8;
  end;
