
theorem
  for S1,S2 being non empty reflexive RelStr for D being non empty
  directed Subset of [:S1,S2:] holds [:proj1 D,proj2 D:] c= downarrow D
proof
  let S1,S2 be non empty reflexive RelStr, D be non empty directed Subset of
  [:S1,S2:];
  reconsider C1 = the carrier of S1, C2 = the carrier of S2 as non empty set;
  let q be object;
  reconsider D9 = D as non empty Subset of [:C1,C2:] by Def2;
  set D1 = proj1 D, D2 = proj2 D;
A1: downarrow D = {x where x is Element of [:S1,S2:] : ex y being Element of
  [:S1,S2:] st x <= y & y in D} by WAYBEL_0:14;
A2: D9 c= [:D1,D2:] by Th1;
  proj2 D9 is non empty;
  then reconsider D2 as non empty Subset of S2;
  proj1 D9 is non empty;
  then reconsider D1 as non empty Subset of S1;
  assume q in [:proj1 D,proj2 D:];
  then consider d, e being object such that
A3: d in D1 and
A4: e in D2 and
A5: q = [d,e] by ZFMISC_1:def 2;
  consider y being object such that
A6: [d,y] in D by A3,XTUPLE_0:def 12;
  consider x being object such that
A7: [x,e] in D by A4,XTUPLE_0:def 13;
  reconsider y, e as Element of D2 by A6,A7,XTUPLE_0:def 13;
  reconsider x, d as Element of D1 by A6,A7,XTUPLE_0:def 12;
  consider z being Element of [:S1,S2:] such that
A8: z in D and
A9: [d,y] <= z & [x,e] <= z by A6,A7,WAYBEL_0:def 1;
  consider z1, z2 being object such that
A10: z1 in D1 and
A11: z2 in D2 and
A12: z = [z1,z2] by A2,A8,ZFMISC_1:def 2;
  reconsider z2 as Element of D2 by A11;
  reconsider z1 as Element of D1 by A10;
  d <= z1 & e <= z2 by A9,A10,A11,A12,Th11;
  then [d,e] <= [z1,z2] by Th11;
  hence thesis by A5,A1,A8,A12;
end;
