
theorem
  for S, T being antisymmetric up-complete non empty reflexive RelStr,
  X being Subset of [:S,T:] st X is property(S) holds proj1 X is property(S) &
  proj2 X is property(S)
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 ex y
being Element of [:S,T:] st y in D & for x being Element of [:S,T:] st x in D &
  x >= y holds x in 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;
    reconsider t as Element of T by A2,A3,ZFMISC_1:87;
    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 YELLOW_0:39;
    then consider y being Element of [:S,T:] such that
A4: y in [:D,t9:] and
A5: for x being Element of [:S,T:] st x in [:D,t9:] & x >= y holds x
    in X by A1,A3;
    take z = y`1;
A6: y = [y`1,y`2] by A2,MCART_1:21;
    hence z in D by A4,ZFMISC_1:87;
A7: y`2 = t by A4,A6,ZFMISC_1:106;
    let x be Element of S;
    assume x in D;
    then
A8: [x,t] in [:D,t9:] by ZFMISC_1:106;
A9: y`2 <= y`2;
    assume x >= z;
    then [x,t] >= y by A6,A7,A9,YELLOW_3:11;
    then [x,t] in X by A5,A8;
    hence x in proj1 X by XTUPLE_0:def 12;
  end;
  let D be non empty directed Subset of T;
  assume sup D in proj2 X;
  then consider s being object such that
A10: [s,sup D] in X by XTUPLE_0:def 13;
  reconsider s as Element of S by A2,A10,ZFMISC_1:87;
  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 YELLOW_0:39;
  then consider y being Element of [:S,T:] such that
A11: y in [:s9,D:] and
A12: for x being Element of [:S,T:] st x in [:s9,D:] & x >= y holds x in
  X by A1,A10;
  take z = y`2;
A13: y = [y`1,y`2] by A2,MCART_1:21;
  hence z in D by A11,ZFMISC_1:87;
A14: y`1 = s by A11,A13,ZFMISC_1:105;
  let x be Element of T;
  assume x in D;
  then
A15: [s,x] in [:s9,D:] by ZFMISC_1:105;
A16: y`1 <= y`1;
  assume x >= z;
  then [s,x] >= y by A13,A14,A16,YELLOW_3:11;
  then [s,x] in X by A12,A15;
  hence thesis by XTUPLE_0:def 13;
end;
