let S be non empty up-complete Poset; :: thesis: for T being non empty lower-bounded up-complete Poset
for x being Element of [:S,T:] holds proj1 (waybelow x) = waybelow (x `1)
let T be non empty lower-bounded up-complete Poset; :: thesis: for x being Element of [:S,T:] holds proj1 (waybelow x) = waybelow (x `1)
let x be Element of [:S,T:]; :: thesis: proj1 (waybelow x) = waybelow (x `1)
A1: Bottom T << x `2 by WAYBEL_3:4;
thus proj1 (waybelow x) c= waybelow (x `1) by Th45; :: according to XBOOLE_0:def 10 :: thesis: waybelow (x `1) c= proj1 (waybelow x)
let a be object ; :: according to TARSKI:def 3 :: thesis: ( not a in waybelow (x `1) or a in proj1 (waybelow x) )
assume A2: a in waybelow (x `1) ; :: thesis: a in proj1 (waybelow x)
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 a in proj1 (waybelow x) by A3, XTUPLE_0:def 12; :: thesis: verum