
theorem
  for S, T being up-complete non empty Poset, X being Subset of S, Y
  being Subset of T st X is property(S) & Y is property(S) holds [:X,Y:] is
  property(S)
proof
  let S, T be up-complete non empty Poset, X be Subset of S, Y be Subset of
  T such that
A1: for D being non empty directed Subset of S st sup D in X ex y being
Element of S st y in D & for x being Element of S st x in D & x >= y holds x in
  X and
A2: for D being non empty directed Subset of T st sup D in Y ex y being
Element of T st y in D & for x being Element of T st x in D & x >= y holds x in
  Y;
  let D be non empty directed Subset of [:S,T:] such that
A3: sup D in [:X,Y:];
  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 & sup proj1 D in X by A3,YELLOW_3:21,22
,ZFMISC_1:87;
  then consider s being Element of S such that
A5: s in proj1 D and
A6: for x being Element of S st x in proj1 D & x >= s holds x in X by A1;
  consider s2 being object such that
A7: [s,s2] in D by A5,XTUPLE_0:def 12;
  proj2 D is non empty directed & sup proj2 D in Y by A3,A4,YELLOW_3:21,22
,ZFMISC_1:87;
  then consider t being Element of T such that
A8: t in proj2 D and
A9: for x being Element of T st x in proj2 D & x >= t holds x in Y by A2;
  consider t1 being object such that
A10: [t1,t] 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 s2 as Element of T by A7,ZFMISC_1:87;
  reconsider t1 as Element of S by A11,A10,ZFMISC_1:87;
  consider z being Element of [:S,T:] such that
A12: z in D and
A13: [s,s2] <= z and
A14: [t1,t] <= z by A7,A10,WAYBEL_0:def 1;
A15: z = [z`1,z`2] by A11,MCART_1:21;
  then
A16: t <= z`2 by A14,YELLOW_3:11;
  take z;
  thus z in D by A12;
  let x be Element of [:S,T:] such that
A17: x in D;
  assume
A18: x >= z;
  then
A19: x`2 >= z`2 by YELLOW_3:12;
A20: x = [x`1,x`2] by A11,MCART_1:21;
  then x`2 in proj2 D by A17,XTUPLE_0:def 13;
  then
A21: x`2 in Y by A9,A19,A16,ORDERS_2:3;
A22: s <= z`1 by A13,A15,YELLOW_3:11;
A23: x`1 >= z`1 by A18,YELLOW_3:12;
  x`1 in proj1 D by A17,A20,XTUPLE_0:def 12;
  then x`1 in X by A6,A23,A22,ORDERS_2:3;
  hence thesis by A20,A21,ZFMISC_1:87;
end;
