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 Th3:
  iterSet(g,Bottom P) is non empty Chain of P
  proof
  set a = Bottom P;
  set R = the InternalRel of P;
  set L=iterSet(g,a);
  A1:x in L implies x is Element of P
    proof
    assume x in L;
    then consider y being Element of P such that
A2: x=y & ex n being Nat st y = iter(g,n).a;
    thus thesis by A2;
    end;
   for x being object holds x in L implies x in the carrier of P
    proof let x be object;
    assume x in L;
    then x is Element of the carrier of P by A1;
    hence thesis;
    end;
  then reconsider L as Subset of P by TARSKI:def 3;
  for x,y being object
holds x in L & y in L & x <>y implies [x,y] in R  or [y,x] in R
    proof let x,y be object;
    assume A3:x in L & y in L & x <>y;
    then reconsider x,y as Element of P;
    consider p such that A4: x=p & ex n st p = iter(g,n).a by A3;
    consider p1 such that A5: y=p1 & ex m st p1 = iter(g,m).a by A3;
    consider n such that A6:p = iter(g,n).a by A4;
    consider m such that A7:p1 = iter(g,m).a by A5;
    p<=p1 or p1<=p
      proof
      set a1 = iter(g,1).a;
      A8:a1=g.a by FUNCT_7:70;
      per cases;
        suppose n<=m;
          hence thesis by A6,A7,A8,Lm5,YELLOW_0:44;
          end;
        suppose m<=n;
          hence thesis by A6,A7,A8,Lm5,YELLOW_0:44;
          end;
      end;
    hence thesis by A4,A5,ORDERS_2:def 5;
    end;
  then A9:R is_connected_in L by RELAT_2:def 6;
  for x being object holds x in L implies [x,x] in R
    proof let x be object;
    assume x in L;
    then reconsider x as Element of P;
    x<=x;
    hence thesis by ORDERS_2:def 5;
    end;
  then R is_reflexive_in L by RELAT_2:def 1;
  then R is_strongly_connected_in L by A9,ORDERS_1:7;
  then reconsider L as Chain of P by ORDERS_2:def 7;
  L is non empty Chain of P;
  hence thesis;
end;
