let S be non empty lower-bounded up-complete Poset; :: thesis: for T being non empty up-complete Poset
for x being Element of [:S,T:] holds proj2 (compactbelow x) = compactbelow (x `2 )
let T be non empty up-complete Poset; :: thesis: for x being Element of [:S,T:] holds proj2 (compactbelow x) = compactbelow (x `2 )
let x be Element of [:S,T:]; :: thesis: proj2 (compactbelow x) = compactbelow (x `2 )
the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by YELLOW_3:def 2;
then A1: x = [(x `1 ),(x `2 )] by MCART_1:23;
thus proj2 (compactbelow x) c= compactbelow (x `2 ) by Th51; :: according to XBOOLE_0:def 10 :: thesis: compactbelow (x `2 ) c= proj2 (compactbelow x)
let a be set ; :: according to TARSKI:def 3 :: thesis: ( not a in compactbelow (x `2 ) or a in proj2 (compactbelow x) )
assume A2: a in compactbelow (x `2 ) ; :: thesis: a in proj2 (compactbelow x)
then reconsider a' = a as Element of T ;
A3: ( a' <= x `2 & a' is compact ) by A2, WAYBEL_8:4;
Bottom S <= x `1 by YELLOW_0:44;
then A4: [(Bottom S),a'] <= [(x `1 ),(x `2 )] by A3, YELLOW_3:11;
( [(Bottom S),a'] `1 = Bottom S & [(Bottom S),a'] `2 = a' ) by MCART_1:7;
then [(Bottom S),a'] is compact by A3, Th23, WAYBEL_3:15;
then [(Bottom S),a'] in compactbelow [(x `1 ),(x `2 )] by A4;
hence a in proj2 (compactbelow x) by A1, RELAT_1:def 5; :: thesis: verum