
theorem Th69:
  for S, T being lower-bounded non empty Poset st [:S,T:] is
  algebraic holds S is algebraic & T is algebraic
proof
  let S, T be lower-bounded non empty Poset such that
A1: for x being Element of [:S,T:] holds compactbelow x is non empty
  directed and
A2: [:S,T:] is up-complete satisfying_axiom_K;
A3: S is up-complete & T is up-complete by A2,WAYBEL_2:11;
  hereby
    hereby
      set t = the Element of T;
      let s be Element of S;
A4:   compactbelow [s,t] is directed by A1;
      [:compactbelow s,compactbelow t:] = compactbelow [s,t] & proj1 [:
      compactbelow s,compactbelow t:] = compactbelow s by A3,Th50,FUNCT_5:9;
      hence compactbelow s is non empty directed by A4,YELLOW_3:22;
    end;
    thus S is up-complete by A2,WAYBEL_2:11;
    thus S is satisfying_axiom_K
    proof
      set t = the Element of T;
      let s be Element of S;
      compactbelow [s,t] is non empty directed by A1;
      then ex_sup_of compactbelow [s,t], [:S,T:] by A2,WAYBEL_0:75;
      then
A5:   sup compactbelow [s,t] = [sup proj1 compactbelow [s,t],sup proj2
      compactbelow [s,t]] by Th5;
      thus s = [s,t]`1
        .= (sup compactbelow [s,t])`1 by A2
        .= sup proj1 compactbelow [s,t] by A5
        .= sup compactbelow [s,t]`1 by A3,Th52
        .= sup compactbelow s;
    end;
  end;
  set s = the Element of S;
  hereby
    set s = the Element of S;
    let t be Element of T;
A6: compactbelow [s,t] is directed by A1;
    [:compactbelow s,compactbelow t:] = compactbelow [s,t] & proj2 [:
    compactbelow s,compactbelow t:] = compactbelow t by A3,Th50,FUNCT_5:9;
    hence compactbelow t is non empty directed by A6,YELLOW_3:22;
  end;
  thus T is up-complete by A2,WAYBEL_2:11;
  let t be Element of T;
  compactbelow [s,t] is non empty directed by A1;
  then ex_sup_of compactbelow [s,t], [:S,T:] by A2,WAYBEL_0:75;
  then
A7: sup compactbelow [s,t] = [sup proj1 compactbelow [s,t],sup proj2
  compactbelow [s,t]] by Th5;
  thus t = [s,t]`2
    .= (sup compactbelow [s,t])`2 by A2
    .= sup proj2 compactbelow [s,t] by A7
    .= sup compactbelow [s,t]`2 by A3,Th53
    .= sup compactbelow t;
end;
