
theorem Th12:
  for L being up-complete antisymmetric non empty reflexive
  RelStr for D being non empty directed Subset of [:L,L:] holds sup D = [sup
  proj1 D,sup proj2 D]
proof
  let L be up-complete antisymmetric non empty reflexive RelStr, D be non
  empty directed Subset of [:L,L:];
  reconsider D1 = proj1 D, D2 = proj2 D as non empty directed Subset of L by
YELLOW_3:21,22;
  reconsider C = the carrier of L as non empty set;
  reconsider D9 = D as non empty Subset of [:C,C:] by YELLOW_3:def 2;
A1: ex_sup_of D1,L by WAYBEL_0:75;
  the carrier of [:L,L:] = [:C,C:] by YELLOW_3:def 2;
  then consider d1, d2 being object such that
A2: d1 in C & d2 in C and
A3: sup D = [d1,d2] by ZFMISC_1:def 2;
A4: ex_sup_of D2,L by WAYBEL_0:75;
  reconsider d1, d2 as Element of L by A2;
A5: ex_sup_of D,[:L,L:] by WAYBEL_0:75;
  D2 is_<=_than d2
  proof
    let b be Element of L;
    assume b in D2;
    then consider x being object such that
A6: [x,b] in D by XTUPLE_0:def 13;
    reconsider x as Element of D1 by A6,XTUPLE_0:def 12;
    D is_<=_than [d1,d2] by A5,A3,YELLOW_0:def 9;
    then [x,b] <= [d1,d2] by A6;
    hence b <= d2 by YELLOW_3:11;
  end;
  then
A7: sup D2 <= d2 by A4,YELLOW_0:def 9;
  D1 is_<=_than d1
  proof
    let b be Element of L;
    assume b in D1;
    then consider x being object such that
A8: [b,x] in D by XTUPLE_0:def 12;
    reconsider x as Element of D2 by A8,XTUPLE_0:def 13;
    D is_<=_than [d1,d2] by A5,A3,YELLOW_0:def 9;
    then [b,x] <= [d1,d2] by A8;
    hence b <= d1 by YELLOW_3:11;
  end;
  then sup D1 <= d1 by A1,YELLOW_0:def 9;
  then
A9: [sup D1,sup D2] <= sup D by A3,A7,YELLOW_3:11;
A10: ex_sup_of [:D1,D2:],[:L,L:] by WAYBEL_0:75;
  reconsider D1, D2 as non empty Subset of L;
  D9 c= [:D1,D2:] by YELLOW_3:1;
  then sup D <= sup [:D1,D2:] by A5,A10,YELLOW_0:34;
  then sup D <= [sup proj1 D,sup proj2 D] by A1,A4,YELLOW_3:43;
  hence thesis by A9,ORDERS_2:2;
end;
