
theorem Th39:
  for S1, S2 being antisymmetric non empty RelStr for D1 being non
  empty Subset of S1, D2 being non empty Subset of S2 holds ex_sup_of D1,S1 &
  ex_sup_of D2,S2 iff ex_sup_of [:D1,D2:],[:S1,S2:]
proof
  let S1, S2 be antisymmetric non empty RelStr, D1 be non empty Subset of S1,
  D2 be non empty Subset of S2;
  hereby
    assume that
A1: ex_sup_of D1,S1 and
A2: ex_sup_of D2,S2;
    consider a2 being Element of S2 such that
A3: D2 is_<=_than a2 and
A4: for b being Element of S2 st D2 is_<=_than b holds a2 <= b by A2,
YELLOW_0:15;
    consider a1 being Element of S1 such that
A5: D1 is_<=_than a1 and
A6: for b being Element of S1 st D1 is_<=_than b holds a1 <= b by A1,
YELLOW_0:15;
    ex a being Element of [:S1,S2:] st [:D1,D2:] is_<=_than a & for b
    being Element of [:S1,S2:] st [:D1,D2:] is_<=_than b holds a <= b
    proof
      take a = [a1,a2];
      thus [:D1,D2:] is_<=_than a by A5,A3,Th30;
      let b be Element of [:S1,S2:] such that
A7:   [:D1,D2:] is_<=_than b;
      the carrier of [:S1,S2:] = [:the carrier of S1, the carrier of S2 :]
      by Def2;
      then
A8:   b = [b`1,b`2] by MCART_1:21;
      then D2 is_<=_than b`2 by A7,Th29;
      then
A9:   a2 <= b`2 by A4;
      D1 is_<=_than b`1 by A7,A8,Th29;
      then a1 <= b`1 by A6;
      hence thesis by A8,A9,Th11;
    end;
    hence ex_sup_of [:D1,D2:],[:S1,S2:] by YELLOW_0:15;
  end;
  assume ex_sup_of [:D1,D2:],[:S1,S2:];
  then consider x being Element of [:S1,S2:] such that
A10: [:D1,D2:] is_<=_than x and
A11: for b being Element of [:S1,S2:] st [:D1,D2:] is_<=_than b holds x
  <= b by YELLOW_0:15;
  the carrier of [:S1,S2:] = [:the carrier of S1,the carrier of S2:] by Def2;
  then
A12: x = [x`1,x`2] by MCART_1:21;
  then
A13: D1 is_<=_than x`1 by A10,Th29;
A14: D2 is_<=_than x`2 by A10,A12,Th29;
  for b being Element of S1 st D1 is_<=_than b holds x`1 <= b
  proof
    let b be Element of S1;
    assume D1 is_<=_than b;
    then x <= [b,x`2] by A11,A14,Th30;
    hence thesis by A12,Th11;
  end;
  hence ex_sup_of D1,S1 by A13,YELLOW_0:15;
  for b being Element of S2 st D2 is_<=_than b holds x`2 <= b
  proof
    let b be Element of S2;
    assume D2 is_<=_than b;
    then x <= [x`1,b] by A11,A13,Th30;
    hence thesis by A12,Th11;
  end;
  hence thesis by A14,YELLOW_0:15;
end;
