
theorem Th43:
  for S1, S2 being antisymmetric non empty RelStr for D1 being non
  empty Subset of S1, D2 being non empty Subset of S2 st ex_sup_of D1,S1 &
  ex_sup_of D2,S2 holds sup [:D1,D2:] = [sup D1,sup D2]
proof
  let S1, S2 be antisymmetric non empty RelStr, D1 be non empty Subset of S1,
  D2 be non empty Subset of S2 such that
A1: ex_sup_of D1,S1 & ex_sup_of D2,S2;
  set s = sup [:D1,D2:];
  s is Element of [:the carrier of S1,the carrier of S2:] by Def2;
  then consider s1, s2 being object such that
A2: s1 in the carrier of S1 and
A3: s2 in the carrier of S2 and
A4: s = [s1,s2] by ZFMISC_1:def 2;
  reconsider s2 as Element of S2 by A3;
  reconsider s1 as Element of S1 by A2;
A5: ex_sup_of [:D1,D2:],[:S1,S2:] by A1,Th39;
  then
A6: [s1,s2] is_>=_than [:D1,D2:] by A4,YELLOW_0:30;
  then
A7: s1 is_>=_than D1 by Th29;
A8: for b being Element of [:S1,S2:] st b is_>=_than [:D1,D2:] holds [s1,s2]
  <= b by A4,A5,YELLOW_0:30;
  then for b being Element of S1 st b is_>=_than D1 holds s1 <= b by A1,Th35;
  then
A9: s1 = sup D1 by A7,YELLOW_0:30;
A10: s2 is_>=_than D2 by A6,Th29;
  for b being Element of S2 st b is_>=_than D2 holds s2 <= b by A1,A8,Th35;
  hence thesis by A4,A9,A10,YELLOW_0:30;
end;
