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 Th2:
  sup (f.:L) <= f.(sup L)
  proof
  reconsider M=f.:L as non empty Chain of Q by Th1;
  set r = sup L;
  set u = f.(sup L);
  A1:ex_sup_of L,P & ex_sup_of M,Q by Def1;
  for q st q in M holds q <= u
    proof
    let q;
    assume q in M;
    then consider x being object such that
A2: x in dom f & x in L & q = f.x by FUNCT_1:def 6;
    reconsider x as Element of P by A2;
    L is_<=_than r by A1,YELLOW_0:def 9;
    then x <= r by A2;
    hence thesis by A2,ORDERS_3:def 5;
    end;
  then M is_<=_than u;
  hence thesis by A1,YELLOW_0:def 9;
  end;
