reserve n,m,k for Nat;
reserve x,y,z,X for set;
reserve P,Q for strict chain-complete non empty Poset;
reserve L for non empty Chain of P;
reserve M for non empty Chain of Q;
reserve p,p1,p2,p3,p4 for Element of P;
reserve q,q1,q2 for Element of Q;
reserve f for monotone Function of P,Q;
reserve g,g1,g2 for monotone Function of P,P;

theorem Th8:
  for f being monotone Function of P,Q holds
    (for L holds f.(sup L) <= sup (f.:L)) implies f is continuous
  proof
  let f be monotone Function of P,Q;
  assume A1:for L holds f.(sup L) <= sup (f.:L);
  for L holds f.(sup L) = sup (f.:L)
    proof
    let L;
    set a1=f.(sup L);
    set a2=sup (f.:L);
    A2:a2<=a1 by Th2;
    a1<=a2 by A1;
    hence thesis by A2,ORDERS_2:2;
    end;
  hence thesis by Th6;
  end;
