
theorem Th44:
  for S, T being up-complete non empty Poset, s being Element of
  S, t being Element of T holds [:waybelow s,waybelow t:] = waybelow [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 [:waybelow s,waybelow t:];
    then consider x1, x2 being object such that
A1: x1 in waybelow s and
A2: x2 in waybelow 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_3:7;
    then [s,t] >> [x1,x2] by Th19;
    hence x in waybelow [s,t] by A3;
  end;
  let x be object;
  assume
A4: x in waybelow [s,t];
  then reconsider x9 = x as Element of [:S,T:];
  the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
YELLOW_3:def 2;
  then
A5: x9 = [x9`1,x9`2] by MCART_1:21;
A6: [s,t] >> x9 by A4,WAYBEL_3:7;
  then t >> x9`2 by A5,Th19;
  then
A7: x`2 in waybelow t;
  s >> x9`1 by A5,A6,Th19;
  then x`1 in waybelow s;
  hence thesis by A5,A7,ZFMISC_1:def 2;
end;
