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 Th4:
  sup iter_min(g) = sup (g.:iter_min(g))
  proof
  reconsider L=g.:iter_min(g) as non empty Chain of P by Th1;
  A1:ex_sup_of iter_min(g),P & ex_sup_of L,P by Def1;
  set a = Bottom P;
  set s1=sup iter_min(g);
  set s2=sup L;
  A2:iter_min(g) is_<=_than s1 by A1,YELLOW_0:def 9;
  A3:L is_<=_than s2 by A1,YELLOW_0:def 9;
  for x being Element of P st x in iter_min(g) holds x <= s2
    proof
    let x be Element of P;
    assume x in iter_min(g);
    then consider p such that A4:x=p & ex n st p = iter(g,n).a;
    consider n such that A5:p=iter(g,n).a by A4;
    A6:1<=n implies p in L
      proof
      assume 1<=n;
      then consider k such that A7:n=1+k by NAT_1:10;
      reconsider z=iter(g,k).a as Element of P;
      z in the carrier of P; then
A8:   z in dom g & z in iter_min(g) by FUNCT_2:def 1;
      p = (g*iter(g,k)).a by A5,A7,FUNCT_7:71
       .= g.z by Lm2;
      hence thesis by A8,FUNCT_1:def 6;
      end;
    n=0 implies p=a by A5,Lm1;
    hence thesis by A6,A4,A3,NAT_1:14,YELLOW_0:44;
    end;
  then iter_min(g) is_<=_than s2;
  then A9:s1<=s2 by A1,YELLOW_0:30;
  for x being Element of P st x in L holds x <= s1
    proof
    let x be Element of P;
    assume x in L;
    then consider z being object such that
A10: z in dom g & z in iter_min(g) & x = g.z by FUNCT_1:def 6;
    consider z1 be Element of P such that
               A11:z=z1 & ex n st z1 = iter(g,n).a by A10;
    consider n such that A12:z1=iter(g,n).a by A11;
    set n1=n+1;
    g.z = (g*iter(g,n)).a by Lm2,A11,A12
       .= iter(g,n1).a by FUNCT_7:71;
    then x in iterSet(g,a) by A10;
    hence thesis by A2;
    end;
  then L is_<=_than s1;
  then s2<=s1 by A1,YELLOW_0:def 9;
  hence thesis by A9,ORDERS_2:2;
  end;
