
theorem
  for S being reflexive RelStr, T being RelStr, X being Subset of [:S,T
  :] holds proj2 downarrow X = downarrow proj2 X
proof
  let S be reflexive RelStr, T be RelStr, X be Subset of [:S,T:];
  thus proj2 downarrow X c= downarrow proj2 X by Th28;
  let c be object;
  assume
A1: c in downarrow proj2 X;
  then reconsider T9 = T as non empty RelStr;
  reconsider c9 = c as Element of T9 by A1;
  consider b being Element of T9 such that
A2: b >= c9 and
A3: b in proj2 X by A1,WAYBEL_0:def 15;
  consider b1 being object such that
A4: [b1,b] in X by A3,XTUPLE_0:def 13;
A5: the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
YELLOW_3:def 2;
  then reconsider S9 = S as non empty reflexive RelStr by A4,ZFMISC_1:87;
  reconsider b1 as Element of S9 by A5,A4,ZFMISC_1:87;
  b1 <= b1;
  then [b1,b] >= [b1,c9] by A2,YELLOW_3:11;
  then [b1,c9] in downarrow X by A4,WAYBEL_0:def 15;
  hence thesis by XTUPLE_0:def 13;
end;
