
theorem Th52:
  for S being up-complete non empty Poset, T being up-complete
  lower-bounded non empty Poset, x being Element of [:S,T:] holds proj1
  compactbelow x = compactbelow x`1
proof
  let S be up-complete non empty Poset, T be up-complete lower-bounded non
  empty Poset, x be Element of [:S,T:];
A1: Bottom T <= x`2 by YELLOW_0:44;
  thus proj1 compactbelow x c= compactbelow x`1 by Th51;
  let a be object;
  assume
A2: a in compactbelow x`1;
  then reconsider a9 = a as Element of S;
  a9 <= x`1 by A2,WAYBEL_8:4;
  then
A3: [a9,Bottom T] <= [x`1,x`2] by A1,YELLOW_3:11;
  the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
YELLOW_3:def 2;
  then
A4: x = [x`1,x`2] by MCART_1:21;
A5: [a9,Bottom T]`1 = a9 & [a9,Bottom T]`2 = Bottom T;
  a9 is compact by A2,WAYBEL_8:4;
  then [a9,Bottom T] is compact by A5,Th23,WAYBEL_3:15;
  then [a9,Bottom T] in compactbelow [x`1,x`2] by A3;
  hence thesis by A4,XTUPLE_0:def 12;
end;
