
theorem
  for S, T being antisymmetric up-complete non empty reflexive RelStr,
X being Subset of [:S,T:] st X is inaccessible holds proj1 X is inaccessible &
  proj2 X is inaccessible
proof
  let S, T be antisymmetric up-complete non empty reflexive RelStr, X be
  Subset of [:S,T:] such that
A1: for D being non empty directed Subset of [:S,T:] st sup D in X holds
  D meets X;
A2: the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
YELLOW_3:def 2;
  hereby
    let D be non empty directed Subset of S;
    assume sup D in proj1 X;
    then consider t being object such that
A3: [sup D, t] in X by XTUPLE_0:def 12;
A4: t in the carrier of T by A2,A3,ZFMISC_1:87;
    then reconsider t9 = {t} as non empty directed Subset of T by WAYBEL_0:5;
    ex_sup_of [:D,t9:],[:S,T:] by WAYBEL_0:75;
    then sup [:D,t9:] = [sup proj1 [:D,t9:], sup proj2 [:D,t9:]] by YELLOW_3:46
      .= [sup D, sup proj2 [:D,t9:]] by FUNCT_5:9
      .= [sup D,sup t9] by FUNCT_5:9
      .= [sup D,t] by A4,YELLOW_0:39;
    then [:D,{t}:] meets X by A1,A3;
    then consider x being object such that
A5: x in [:D,{t}:] and
A6: x in X by XBOOLE_0:3;
    now
      take a = x`1;
      x = [a,x`2] by A5,MCART_1:21;
      hence a in D & a in proj1 X by A5,A6,XTUPLE_0:def 12,ZFMISC_1:87;
    end;
    hence D meets proj1 X by XBOOLE_0:3;
  end;
  let D be non empty directed Subset of T;
  assume sup D in proj2 X;
  then consider s being object such that
A7: [s,sup D] in X by XTUPLE_0:def 13;
A8: s in the carrier of S by A2,A7,ZFMISC_1:87;
  then reconsider s9 = {s} as non empty directed Subset of S by WAYBEL_0:5;
  ex_sup_of [:s9,D:],[:S,T:] by WAYBEL_0:75;
  then sup [:s9,D:] = [sup proj1 [:s9,D:], sup proj2 [:s9,D:]] by YELLOW_3:46
    .= [sup s9, sup proj2 [:s9,D:]] by FUNCT_5:9
    .= [sup s9,sup D] by FUNCT_5:9
    .= [s,sup D] by A8,YELLOW_0:39;
  then [:{s},D:] meets X by A1,A7;
  then consider x being object such that
A9: x in [:{s},D:] and
A10: x in X by XBOOLE_0:3;
  now
    take a = x`2;
    x = [x`1,a] by A9,MCART_1:21;
    hence a in D & a in proj2 X by A9,A10,XTUPLE_0:def 13,ZFMISC_1:87;
  end;
  hence thesis by XBOOLE_0:3;
end;
