
theorem
  for L1,L2 being antisymmetric non empty RelStr for D being non empty
Subset of [:L1,L2:] st [:L1,L2:] is complete or ex_sup_of D,[:L1,L2:] holds sup
  D = [sup proj1 D,sup proj2 D]
proof
  let L1,L2 be antisymmetric non empty RelStr, D be non empty Subset of [:L1,
  L2:];
  reconsider C1 = the carrier of L1, C2 = the carrier of L2 as non empty set;
  the carrier of [:L1,L2:] = [:C1,C2:] by Def2;
  then consider d1, d2 being object such that
A1: d1 in C1 and
A2: d2 in C2 and
A3: sup D = [d1,d2] by ZFMISC_1:def 2;
  reconsider d1 as Element of L1 by A1;
  reconsider D9 = D as non empty Subset of [:C1,C2:] by Def2;
  proj1 D9 is non empty;
  then reconsider D1=proj1 D as non empty Subset of L1;
  proj2 D9 is non empty;
  then reconsider D2=proj2 D as non empty Subset of L2;
A4: D9 c= [:D1,D2:] by Th1;
  reconsider d2 as Element of L2 by A2;
  assume [:L1,L2:] is complete or ex_sup_of D,[:L1,L2:];
  then
A5: ex_sup_of D,[:L1,L2:] by YELLOW_0:17;
  then
A6: ex_sup_of D2,L2 by Th41;
A7: ex_sup_of D1,L1 by A5,Th41;
  then ex_sup_of [:D1,D2:],[:L1,L2:] by A6,Th39;
  then sup D <= sup [:D1,D2:] by A5,A4,YELLOW_0:34;
  then
A8: sup D <= [sup proj1 D,sup proj2 D] by A7,A6,Th43;
  D2 is_<=_than d2
  proof
    let b be Element of L2;
    assume b in D2;
    then consider x being object such that
A9: [x,b] in D by XTUPLE_0:def 13;
    reconsider x as Element of D1 by A9,XTUPLE_0:def 12;
    reconsider x as Element of L1;
    D is_<=_than [d1,d2] by A5,A3,YELLOW_0:def 9;
    then [x,b] <= [d1,d2] by A9;
    hence thesis by Th11;
  end;
  then
A10: sup D2 <= d2 by A6,YELLOW_0:def 9;
  D1 is_<=_than d1
  proof
    let b be Element of L1;
    assume b in D1;
    then consider x being object such that
A11: [b,x] in D by XTUPLE_0:def 12;
    reconsider x as Element of D2 by A11,XTUPLE_0:def 13;
    reconsider x as Element of L2;
    D is_<=_than [d1,d2] by A5,A3,YELLOW_0:def 9;
    then [b,x] <= [d1,d2] by A11;
    hence thesis by Th11;
  end;
  then sup D1 <= d1 by A7,YELLOW_0:def 9;
  then [sup D1,sup D2] <= sup D by A3,A10,Th11;
  hence thesis by A8,ORDERS_2:2;
end;
