
theorem
  for S1, S2 being non empty RelStr for D1 being non empty Subset of S1,
D2 being non empty Subset of S2 st [:D1,D2:] is directed holds D1 is directed &
  D2 is directed
proof
  let S1, S2 be non empty RelStr, D1 be non empty Subset of S1, D2 be non
  empty Subset of S2 such that
A1: [:D1,D2:] is directed;
  thus D1 is directed
  proof
    set q1 = the Element of D2;
    let x, y be Element of S1;
    assume x in D1 & y in D1;
    then [x,q1] in [:D1,D2:] & [y,q1] in [:D1,D2:] by ZFMISC_1:87;
    then consider z being Element of [:S1,S2:] such that
A2: z in [:D1,D2:] and
A3: [x,q1] <= z & [y,q1] <= z by A1;
    reconsider z2 = z`2 as Element of D2 by A2,MCART_1:10;
    reconsider zz = z`1 as Element of D1 by A2,MCART_1:10;
    take zz;
    thus zz in D1;
    ex x,y being object st x in D1 & y in D2 & z = [x,y] by A2,ZFMISC_1:def 2;
    then [x,q1] <= [zz,z2] & [y,q1] <= [zz,z2] by A3;
    hence thesis by Th11;
  end;
  set q1 = the Element of D1;
  let x, y be Element of S2;
  assume x in D2 & y in D2;
  then [q1,x] in [:D1,D2:] & [q1,y] in [:D1,D2:] by ZFMISC_1:87;
  then consider z being Element of [:S1,S2:] such that
A4: z in [:D1,D2:] and
A5: [q1,x] <= z & [q1,y] <= z by A1;
  reconsider z2 = z`1 as Element of D1 by A4,MCART_1:10;
  reconsider zz = z`2 as Element of D2 by A4,MCART_1:10;
  take zz;
  thus zz in D2;
  ex x,y being object st x in D1 & y in D2 & z = [x,y] by A4,ZFMISC_1:def 2;
  then [q1,x] <= [z2,zz] & [q1,y] <= [z2,zz] by A5;
  hence thesis by Th11;
end;
