
theorem Th19:
  for S, T being up-complete non empty Poset for a, c being
Element of S, b, d being Element of T holds [a,b] << [c,d] iff a << c & b << d
proof
  let S, T be up-complete non empty Poset, a, c be Element of S, b, d be
  Element of T;
  thus [a,b] << [c,d] implies a << c & b << d by Th18;
  assume
A1: for D being non empty directed Subset of S st c <= sup D ex e being
  Element of S st e in D & a <= e;
  assume
A2: for D being non empty directed Subset of T st d <= sup D ex e being
  Element of T st e in D & b <= e;
  let D be non empty directed Subset of [:S,T:] such that
A3: [c,d] <= sup D;
  ex_sup_of D,[:S,T:] by WAYBEL_0:75;
  then
A4: sup D = [sup proj1 D,sup proj2 D] by YELLOW_3:46;
  then proj1 D is non empty directed & c <= sup proj1 D by A3,YELLOW_3:11,21,22
;
  then consider e being Element of S such that
A5: e in proj1 D and
A6: a <= e by A1;
  consider e2 being object such that
A7: [e,e2] in D by A5,XTUPLE_0:def 12;
  proj2 D is non empty directed & d <= sup proj2 D by A3,A4,YELLOW_3:11,21,22;
  then consider f being Element of T such that
A8: f in proj2 D and
A9: b <= f by A2;
  consider f1 being object such that
A10: [f1,f] in D by A8,XTUPLE_0:def 13;
A11: the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
YELLOW_3:def 2;
  then reconsider e2 as Element of T by A7,ZFMISC_1:87;
  reconsider f1 as Element of S by A11,A10,ZFMISC_1:87;
  consider ef being Element of [:S,T:] such that
A12: ef in D and
A13: [e,e2] <= ef & [f1,f] <= ef by A7,A10,WAYBEL_0:def 1;
A14: ef = [ef`1,ef`2] by A11,MCART_1:21;
  then e <= ef`1 & f <= ef`2 by A13,YELLOW_3:11;
  then
A15: [e,f] <= ef by A14,YELLOW_3:11;
  take ef;
  thus ef in D by A12;
  [a,b] <= [e,f] by A6,A9,YELLOW_3:11;
  hence thesis by A15,ORDERS_2:3;
end;
