
theorem Th51:
  for S, T being antisymmetric up-complete non empty reflexive
RelStr, x being Element of [:S,T:] holds proj1 compactbelow x c= compactbelow
  x`1 & proj2 compactbelow x c= compactbelow 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 compactbelow x;
    then consider b being object such that
A3: [a,b] in compactbelow 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]`1 = a9 & [a9,b] is compact by A3,WAYBEL_8:4;
    then
A4: a9 is compact by Th22;
    [a9,b] <= x by A3,WAYBEL_8:4;
    then a9 <= x`1 by A2,YELLOW_3:11;
    hence a in compactbelow x`1 by A4;
  end;
  let b be object;
  assume b in proj2 compactbelow x;
  then consider a being object such that
A5: [a,b] in compactbelow x by XTUPLE_0:def 13;
  reconsider a as Element of S by A1,A5,ZFMISC_1:87;
  reconsider b9 = b as Element of T by A1,A5,ZFMISC_1:87;
  [a,b9]`2 = b9 & [a,b9] is compact by A5,WAYBEL_8:4;
  then
A6: b9 is compact by Th22;
  [a,b9] <= x by A5,WAYBEL_8:4;
  then b9 <= x`2 by A2,YELLOW_3:11;
  hence thesis by A6;
end;
