
theorem
  for S, T being antisymmetric up-complete non empty reflexive RelStr,
  X being upper Subset of S, Y being upper Subset of T st X is inaccessible & Y
  is inaccessible holds [:X,Y:] is inaccessible
proof
  let S, T be antisymmetric up-complete non empty reflexive RelStr, X be
  upper Subset of S, Y be upper Subset of T such that
A1: for D being non empty directed Subset of S st sup D in X holds D
  meets X and
A2: for D being non empty directed Subset of T st sup D in Y holds D meets 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 proj1 D meets X by A1;
  then consider s being object such that
A5: s in proj1 D and
A6: s in X by XBOOLE_0:3;
  reconsider s as Element of S by A5;
  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 proj2 D meets Y by A2;
  then consider t being object such that
A8: t in proj2 D and
A9: t in Y by XBOOLE_0:3;
  reconsider t as Element of T by A8;
  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;
  now
    take z;
    thus z in D by A12;
A15: z = [z`1,z`2] by A11,MCART_1:21;
    then t <= z`2 by A14,YELLOW_3:11;
    then
A16: z`2 in Y by A9,WAYBEL_0:def 20;
    s <= z`1 by A13,A15,YELLOW_3:11;
    then z`1 in X by A6,WAYBEL_0:def 20;
    hence z in [:X,Y:] by A15,A16,ZFMISC_1:87;
  end;
  hence thesis by XBOOLE_0:3;
end;
