
theorem Th18:
  for S, T being antisymmetric up-complete non empty reflexive
  RelStr, a, c being Element of S, b, d being Element of T st [a,b] << [c,d]
  holds a << c & b << d
proof
  let S, T be antisymmetric up-complete non empty reflexive RelStr, a, c be
  Element of S, b, d be Element of T;
  assume
A1: for D being non empty directed Subset of [:S,T:] st [c,d] <= sup D
  ex e being Element of [:S,T:] st e in D & [a,b] <= e;
A2: the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
YELLOW_3:def 2;
  thus a << c
  proof
    reconsider d9 = {d} as non empty directed Subset of T by WAYBEL_0:5;
    let D be non empty directed Subset of S such that
A3: c <= sup D;
A4: d <= sup d9 by YELLOW_0:39;
    ex_sup_of D,S & ex_sup_of d9,T by WAYBEL_0:75;
    then sup [:D,d9:] = [sup D,sup d9] by YELLOW_3:43;
    then [c,d] <= sup [:D,d9:] by A3,A4,YELLOW_3:11;
    then consider e being Element of [:S,T:] such that
A5: e in [:D,d9:] and
A6: [a,b] <= e by A1;
    take e`1;
    thus e`1 in D by A5,MCART_1:10;
    e = [e`1,e`2] by A2,MCART_1:21;
    hence thesis by A6,YELLOW_3:11;
  end;
  let D be non empty directed Subset of T such that
A7: d <= sup D;
  reconsider c9 = {c} as non empty directed Subset of S by WAYBEL_0:5;
A8: c <= sup c9 by YELLOW_0:39;
  ex_sup_of c9,S & ex_sup_of D,T by WAYBEL_0:75;
  then sup [:c9,D:] = [sup c9,sup D] by YELLOW_3:43;
  then [c,d] <= sup [:c9,D:] by A7,A8,YELLOW_3:11;
  then consider e being Element of [:S,T:] such that
A9: e in [:c9,D:] and
A10: [a,b] <= e by A1;
  take e`2;
  thus e`2 in D by A9,MCART_1:10;
  e = [e`1,e`2] by A2,MCART_1:21;
  hence thesis by A10,YELLOW_3:11;
end;
