
theorem Th45:
  for S, T being antisymmetric up-complete non empty reflexive
  RelStr, x being Element of [:S,T:] holds proj1 waybelow x c= waybelow x`1 &
  proj2 waybelow x c= waybelow x`2
proof
  let S, T be antisymmetric up-complete non empty reflexive RelStr, x be
  Element of [:S,T:];
A1: the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
YELLOW_3:def 2;
  then
A2: x = [x`1,x`2] by MCART_1:21;
  hereby
    let a be object;
    assume a in proj1 waybelow x;
    then consider b being object such that
A3: [a,b] in waybelow x by XTUPLE_0:def 12;
    reconsider b as Element of T by A1,A3,ZFMISC_1:87;
    reconsider a9 = a as Element of S by A1,A3,ZFMISC_1:87;
    [a9,b] << x by A3,WAYBEL_3:7;
    then a9 << x`1 by A2,Th18;
    hence a in waybelow x`1;
  end;
  let b be object;
  assume b in proj2 waybelow x;
  then consider a being object such that
A4: [a,b] in waybelow x by XTUPLE_0:def 13;
  reconsider a as Element of S by A1,A4,ZFMISC_1:87;
  reconsider b9 = b as Element of T by A1,A4,ZFMISC_1:87;
  [a,b9] << x by A4,WAYBEL_3:7;
  then b9 << x`2 by A2,Th18;
  hence thesis;
end;
