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 Th12:
  for f being Element of ConPoset(P,Q) st f = min_func(P,Q)
    holds f is_<=_than the carrier of ConPoset(P,Q)
proof
  let f be Element of ConPoset(P,Q);
  assume A1:f = min_func(P,Q);
  set f1 = min_func(P,Q);
  for x being Element of ConPoset(P,Q) holds f<=x
    proof
    let x be Element of ConPoset(P,Q);
    x in ConFuncs(P,Q);
    then consider x1 being Element of Funcs(the carrier of P,the carrier of Q)
      such that
A2: x = x1 & ex g be continuous Function of P,Q st g=x1;
    consider g being continuous Function of P,Q such that
A3: g=x1 by A2;
    for p holds f1.p <= g.p
      proof
      let p;
      q1=f1.p & q2=g.p implies q1<=q2
        proof
        assume q1=f1.p & q2=g.p;
        then q1=Bottom Q by FUNCOP_1:7;
        hence thesis by YELLOW_0:44;
        end;
      hence thesis;
      end;
    then f1 <= g by YELLOW_2:9;
    then [f,x] in ConRelat(P,Q) by A2,A3,Def7,A1;
    hence thesis by ORDERS_2:def 5;
  end;
  hence thesis;
end;
