
theorem
  for S1, S2 being non empty reflexive RelStr for D being non empty
  lower Subset of [:S1,S2:] holds proj1 D is lower & proj2 D is lower
proof
  let S1, S2 be non empty reflexive RelStr, D be non empty lower Subset of [:
  S1,S2:];
  set D1 = proj1 D, D2 = proj2 D;
  reconsider d1 = D1 as non empty Subset of S1 by Th21;
  the carrier of [:S1,S2:] = [:the carrier of S1, the carrier of S2:] by Def2;
  then
A1: D c= [:D1,D2:] by Th1;
  thus D1 is lower
  proof
    reconsider d2 = D2 as non empty Subset of S2 by Th21;
    let x, y be Element of S1 such that
A2: x in D1 and
A3: x >= y;
    consider q1 being object such that
A4: [x,q1] in D by A2,XTUPLE_0:def 12;
    reconsider q1 as Element of d2 by A1,A4,ZFMISC_1:87;
    q1 <= q1;
    then [x,q1] >= [y,q1] by A3,Th11;
    then [y,q1] in D by A4,WAYBEL_0:def 19;
    hence thesis by A1,ZFMISC_1:87;
  end;
  let x, y be Element of S2 such that
A5: x in D2 and
A6: x >= y;
  consider q1 being object such that
A7: [q1,x] in D by A5,XTUPLE_0:def 13;
  reconsider q1 as Element of d1 by A1,A7,ZFMISC_1:87;
  q1 <= q1;
  then [q1,x] >= [q1,y] by A6,Th11;
  then [q1,y] in D by A7,WAYBEL_0:def 19;
  hence thesis by A1,ZFMISC_1:87;
end;
