
theorem
  for S, T being RelStr, X being Subset of [:S,T:] holds uparrow X c= [:
  uparrow proj1 X,uparrow proj2 X:]
proof
  let S, T be RelStr, X be Subset of [:S,T:];
  let x be object;
  assume
A1: x in uparrow X;
A2: the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
YELLOW_3:def 2;
  then
  ex a, b being object st a in the carrier of S & b in the carrier of T & x =
  [a,b] by A1,ZFMISC_1:def 2;
  then reconsider S9 = S, T9 = T as non empty RelStr;
  reconsider x9 = x as Element of [:S9,T9:] by A1;
  consider y being Element of [:S9,T9:] such that
A3: y <= x9 and
A4: y in X by A1,WAYBEL_0:def 16;
A5: y`1 <= x9`1 by A3,YELLOW_3:12;
A6: y = [y`1,y`2] by A2,MCART_1:21;
  then y`1 in proj1 X by A4,XTUPLE_0:def 12;
  then
A7: x`1 in uparrow proj1 X by A5,WAYBEL_0:def 16;
A8: y`2 <= x9`2 by A3,YELLOW_3:12;
  y`2 in proj2 X by A4,A6,XTUPLE_0:def 13;
  then
A9: x`2 in uparrow proj2 X by A8,WAYBEL_0:def 16;
  x9 = [x9`1,x9`2] by A2,MCART_1:21;
  hence thesis by A7,A9,ZFMISC_1:def 2;
end;
