
theorem Th46:
  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
  waybelow x = waybelow 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 WAYBEL_3:4;
  thus proj1 waybelow x c= waybelow x`1 by Th45;
  let a be object;
  assume
A2: a in waybelow x`1;
  then reconsider a9 = a as Element of S;
  a9 << x`1 by A2,WAYBEL_3:7;
  then [a9,Bottom T] << [x`1,x`2] by A1,Th19;
  then
A3: [a9,Bottom T] 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 12;
end;
