
theorem Th50:
  for S, T being up-complete non empty Poset, s being Element of
S, t being Element of T holds [:compactbelow s,compactbelow t:] = compactbelow
  [s,t]
proof
  let S, T be up-complete non empty Poset, s be Element of S, t be Element
  of T;
  hereby
    let x be object;
    assume x in [:compactbelow s,compactbelow t:];
    then consider x1, x2 being object such that
A1: x1 in compactbelow s and
A2: x2 in compactbelow t and
A3: x = [x1,x2] by ZFMISC_1:def 2;
    reconsider x2 as Element of T by A2;
    reconsider x1 as Element of S by A1;
    s >= x1 & t >= x2 by A1,A2,WAYBEL_8:4;
    then
A4: [s,t] >= [x1,x2] by YELLOW_3:11;
A5: [x1,x2]`1 = x1 & [x1,x2]`2 = x2;
    x1 is compact & x2 is compact by A1,A2,WAYBEL_8:4;
    then [x1,x2] is compact by A5,Th23;
    hence x in compactbelow [s,t] by A3,A4;
  end;
  let x be object;
  assume
A6: x in compactbelow [s,t];
  then reconsider x9 = x as Element of [:S,T:];
A7: x9 is compact by A6,WAYBEL_8:4;
  then
A8: x9`1 is compact by Th22;
A9: x9`2 is compact by A7,Th22;
  the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
YELLOW_3:def 2;
  then
A10: x9 = [x9`1,x9`2] by MCART_1:21;
A11: [s,t] >= x9 by A6,WAYBEL_8:4;
  then t >= x9`2 by A10,YELLOW_3:11;
  then
A12: x`2 in compactbelow t by A9;
  s >= x9`1 by A10,A11,YELLOW_3:11;
  then x`1 in compactbelow s by A8;
  hence thesis by A10,A12,ZFMISC_1:def 2;
end;
