
theorem Th47:
  for S being up-complete lower-bounded non empty Poset, T being
up-complete non empty Poset, x being Element of [:S,T:] holds proj2 waybelow
  x = waybelow x`2
proof
  let S be up-complete lower-bounded non empty Poset, T be up-complete non
  empty Poset, x be Element of [:S,T:];
A1: Bottom S << x`1 by WAYBEL_3:4;
  thus proj2 waybelow x c= waybelow x`2 by Th45;
  let a be object;
  assume
A2: a in waybelow x`2;
  then reconsider a9 = a as Element of T;
  a9 << x`2 by A2,WAYBEL_3:7;
  then [Bottom S,a9] << [x`1,x`2] by A1,Th19;
  then
A3: [Bottom S,a9] in waybelow [x`1,x`2];
  the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
YELLOW_3:def 2;
  then x = [x`1,x`2] by MCART_1:21;
  hence thesis by A3,XTUPLE_0:def 13;
end;
