
theorem
  for S being RelStr, T being reflexive RelStr, X being Subset of [:S,T
  :] holds proj1 uparrow X = uparrow proj1 X
proof
  let S be RelStr, T be reflexive RelStr, X be Subset of [:S,T:];
  thus proj1 uparrow X c= uparrow proj1 X by Th33;
  let a be object;
  assume
A1: a in uparrow proj1 X;
  then reconsider S9 = S as non empty RelStr;
  reconsider a9 = a as Element of S9 by A1;
  consider b being Element of S9 such that
A2: b <= a9 and
A3: b in proj1 X by A1,WAYBEL_0:def 16;
  consider b2 being object such that
A4: [b,b2] in X by A3,XTUPLE_0:def 12;
A5: the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
YELLOW_3:def 2;
  then reconsider T9 = T as non empty reflexive RelStr by A4,ZFMISC_1:87;
  reconsider b2 as Element of T9 by A5,A4,ZFMISC_1:87;
  b2 <= b2;
  then [b,b2] <= [a9,b2] by A2,YELLOW_3:11;
  then [a9,b2] in uparrow X by A4,WAYBEL_0:def 16;
  hence thesis by XTUPLE_0:def 12;
end;
