
theorem
  for S1, S2 being non empty reflexive RelStr for D being non empty
  directed Subset of [:S1,S2:] holds proj1 D is directed & proj2 D is directed
proof
  let S1, S2 be non empty reflexive RelStr, D be non empty directed Subset of
  [:S1,S2:];
  set D1 = proj1 D, D2 = proj2 D;
  the carrier of [:S1,S2:] = [:the carrier of S1, the carrier of S2:] by Def2;
  then
A1: D c= [:proj1 D, proj2 D:] by Th1;
  thus D1 is directed
  proof
    let x, y be Element of S1;
    assume that
A2: x in D1 and
A3: y in D1;
    consider q2 being object such that
A4: [y,q2] in D by A3,XTUPLE_0:def 12;
    reconsider D29 = D2 as non empty Subset of S2 by A2,FUNCT_5:16;
    reconsider D19 = D1 as non empty Subset of S1 by A2;
    consider q1 being object such that
A5: [x,q1] in D by A2,XTUPLE_0:def 12;
    reconsider q2 as Element of D29 by A4,XTUPLE_0:def 13;
    reconsider q1 as Element of D29 by A5,XTUPLE_0:def 13;
    consider z being Element of [:S1,S2:] such that
A6: z in D and
A7: [x,q1] <= z & [y,q2] <= z by A5,A4,WAYBEL_0:def 1;
    reconsider z2 = z`2 as Element of D29 by A1,A6,MCART_1:10;
    reconsider zz = z`1 as Element of D19 by A1,A6,MCART_1:10;
    take zz;
    thus zz in D1;
    ex x,y being object st x in D19 & y in D29 & z = [x,y]
by A1,A6,ZFMISC_1:def 2;
    then z = [zz,z2];
    hence x <= zz & y <= zz by A7,Th11;
  end;
  let x, y be Element of S2;
  assume that
A8: x in D2 and
A9: y in D2;
  consider q2 being object such that
A10: [q2,y] in D by A9,XTUPLE_0:def 13;
  reconsider D29 = D2 as non empty Subset of S2 by A8;
  reconsider D19 = D1 as non empty Subset of S1 by A8,FUNCT_5:16;
  consider q1 being object such that
A11: [q1,x] in D by A8,XTUPLE_0:def 13;
  reconsider q2 as Element of D19 by A10,XTUPLE_0:def 12;
  reconsider q1 as Element of D19 by A11,XTUPLE_0:def 12;
  consider z being Element of [:S1,S2:] such that
A12: z in D and
A13: [q1,x] <= z & [q2,y] <= z by A11,A10,WAYBEL_0:def 1;
  reconsider z2 = z`1 as Element of D19 by A1,A12,MCART_1:10;
  reconsider zz = z`2 as Element of D29 by A1,A12,MCART_1:10;
  take zz;
  thus zz in D2;
  ex x,y being object st x in D19 & y in D29 & z = [x,y]
by A1,A12,ZFMISC_1:def 2;
  then z = [z2,zz];
  hence x <= zz & y <= zz by A13,Th11;
end;
