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 Th5:
  g1 <= g2 implies sup iter_min(g1) <= sup iter_min(g2)
  proof
  assume A1:g1 <= g2;
  set p2 = sup iter_min(g2);
  set a = Bottom P;
  A2:ex_sup_of iter_min(g1),P & ex_sup_of iter_min(g2),P by Def1;
  then A3:iter_min(g2) is_<=_than p2 by YELLOW_0:def 9;
  for x being Element of P st x in iter_min(g1) holds x <= p2
    proof
    let x be Element of P;
    assume x in iter_min(g1);
    then consider p such that A4:x=p & ex n st p = iter(g1,n).a;
    consider n such that A5:p=iter(g1,n).a by A4;
    reconsider y = iter(g2,n).a as Element of P;
    y in iter_min(g2);
    then A6:y <= p2 by A3;
    p <= y by A1,A5,Lm7;
    hence thesis by A4,A6,ORDERS_2:3;
    end;
  then iter_min(g1) is_<=_than p2;
  hence thesis by A2,YELLOW_0:30;
  end;
