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;
reserve F for non empty Chain of ConPoset(P,Q);

theorem Th11:
  ex_sup_of F, ConPoset(P,Q) & sup_func(F) = "\/"(F,ConPoset(P,Q))
  proof
  set X = ConPoset(P,Q);
  set f1 = sup_func(F);
  reconsider f=f1 as Element of ConPoset(P,Q) by Lm21;
  for x being Element of X st x in F holds x <= f
    proof
    let x be Element of X;
    assume A1:x in F;
    then consider g1 being continuous Function of P,Q such that
A2: x = g1 by Lm22;
    reconsider g = g1 as Element of X by A2;
    for p holds g1.p <= f1.p
    proof
      let p;
      q1=g1.p & q2=f1.p implies q1<=q2
      proof
        assume A3:q1=g1.p & q2=f1.p;
        then A4:q1 in F-image p by A1,A2;
        reconsider M = F-image p as non empty Chain of Q;
        q2 = sup M by Def10,A3;
        hence thesis by A4,Lm18;
      end;
      hence thesis;
    end;
    then g1 <= f1 by YELLOW_2:9;
    then [g,f] in ConRelat(P,Q) by Def7;
    hence thesis by A2,ORDERS_2:def 5;
    end;
   then A5:F is_<=_than f;
   for y being Element of X holds F is_<=_than y implies f <= y
    proof
    let y be Element of X;
    y in ConFuncs(P,Q);
    then consider y1 being Element of Funcs(the carrier of P,the carrier of Q)
        such that A6:y = y1 & ex gy be continuous Function of P,Q st gy =y1;
    consider gy being continuous Function of P,Q
        such that A7:gy = y1 by A6;
    F is_<=_than y implies f <= y
      proof
      assume A8:F is_<=_than y;
      for p holds f1.p <= gy.p
        proof
        let p;
        q1=f1.p & q2=gy.p implies q1<=q2
          proof
          assume A9:q1=f1.p & q2=gy.p;
          reconsider M=F-image p as non empty Chain of Q;
          for q st q in M holds q<=q2
            proof
            let q;
            assume q in M;
            then consider a being Element of Q such that
         A10:q=a & ex g being continuous Function of P,Q st g in F & a=g.p;
            consider g be continuous Function of P,Q such that
              A11: g in F & a=g.p by A10;
            reconsider g1 = g as Element of ConPoset(P,Q) by A11;
            g1 <= y by A8,A11;
            then [g1,y] in ConRelat(P,Q) by ORDERS_2:def 5;
            then consider g2,g3 being Function of P,Q such that
         A12:g1=g2 & y=g3 & g2 <= g3 by Def7;
            thus thesis by A12,A6,A7,A10,A11,A9,YELLOW_2:9;
            end;
          then sup M <= q2 by Lm19;
          hence thesis by A9,Def10;
          end;
        hence thesis;
        end;
      then f1 <= gy by YELLOW_2:9;
      then [f,y] in ConRelat(P,Q) by A6,A7,Def7;
      hence thesis by ORDERS_2:def 5;
      end;
    hence thesis;
    end;
  hence thesis by A5,YELLOW_0:30;
  end;
