theorem Th6:
  for f being Function of P,Q holds f is continuous iff
          (f is monotone & for L holds f.(sup L) = sup (f.:L))
   proof
   let f be Function of P,Q;
   thus f is continuous implies
        (f is monotone & for L holds f.(sup L) = sup (f.:L))
     proof
     assume A1:f is continuous;
     for L holds f.(sup L) = sup (f.:L)
     proof
       let L;
    A2:f preserves_sup_of L by A1;
       ex_sup_of L,P by Def1;
       hence thesis by A2,WAYBEL_0:def 31;
     end;
     hence thesis by A1;
     end;
     assume that A3:f is monotone and
A4:  for L holds f.(sup L) = sup (f.:L);
     for L holds f preserves_sup_of L
     proof
       let L;
       reconsider M = f.:L as non empty Chain of Q by A3,Th1;
       ex_sup_of M,Q & f.(sup L) = sup M by Def1,A4;
       hence thesis by WAYBEL_0:def 31;
     end;
     hence thesis by A3;
   end;
