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 Th7:
  for z being Element of Q holds (P-->z) is continuous
  proof
    let z be Element of Q;
    set IT = P --> z;
    for L holds IT.(sup L) = sup (IT.:L)
      proof
      let L;
      set M = IT.:L;
      for x being Element of Q st x in M holds x <= z
        proof
        let x be Element of Q;
        assume x in M;
        then consider a be object such that
    A1: a in dom IT & a in L & x = IT.a by FUNCT_1:def 6;
        thus thesis by A1,FUNCOP_1:7;
        end;
      then A2:M is_<=_than z;
      for y being Element of Q st M is_<=_than y holds z <= y
        proof
        let y be Element of Q;
        assume A3:M is_<=_than y;
        consider a be object such that A4:a in L by XBOOLE_0:def 1;
        a in the carrier of P by A4;
        then A5:a in dom IT by FUNCOP_1:13;
        IT.a = z by A4,FUNCOP_1:7;
        then z in M by A4,A5,FUNCT_1:def 6;
        hence thesis by A3;
        end;
      then z = "\/"(M,Q) by A2,YELLOW_0:30;
      hence thesis by FUNCOP_1:7;
      end;
    hence thesis by Th6;
  end;
