
theorem Th28:
  for S, T being RelStr, X being Subset of [:S,T:] holds proj1
  downarrow X c= downarrow proj1 X & proj2 downarrow X c= downarrow proj2 X
proof
  let S, T be RelStr, X be Subset of [:S,T:];
A1: the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
YELLOW_3:def 2;
  hereby
    let a be object;
    assume a in proj1 downarrow X;
    then consider b being object such that
A2: [a,b] in downarrow X by XTUPLE_0:def 12;
    reconsider S9 = S, T9 = T as non empty RelStr by A1,A2,ZFMISC_1:87;
    reconsider b9 = b as Element of T9 by A1,A2,ZFMISC_1:87;
    reconsider a9 = a as Element of S9 by A1,A2,ZFMISC_1:87;
    consider c being Element of [:S9,T9:] such that
A3: [a9,b9] <= c & c in X by A2,WAYBEL_0:def 15;
    c = [c`1,c`2] by A1,MCART_1:21;
    then a9 <= c`1 & c`1 in proj1 X by A3,XTUPLE_0:def 12,YELLOW_3:11;
    hence a in downarrow proj1 X by WAYBEL_0:def 15;
  end;
  let b be object;
  assume b in proj2 downarrow X;
  then consider a being object such that
A4: [a,b] in downarrow X by XTUPLE_0:def 13;
  reconsider S9 = S, T9 = T as non empty RelStr by A1,A4,ZFMISC_1:87;
  reconsider b9 = b as Element of T9 by A1,A4,ZFMISC_1:87;
  reconsider a9 = a as Element of S9 by A1,A4,ZFMISC_1:87;
  consider c being Element of [:S9,T9:] such that
A5: [a9,b9] <= c & c in X by A4,WAYBEL_0:def 15;
  c = [c`1,c`2] by A1,MCART_1:21;
  then b9 <= c`2 & c`2 in proj2 X by A5,XTUPLE_0:def 13,YELLOW_3:11;
  hence thesis by WAYBEL_0:def 15;
end;
