let S be RelStr ; :: thesis: for T being reflexive RelStr
for X being Subset of [:S,T:] holds proj1 (uparrow X) = uparrow (proj1 X)
let T be reflexive RelStr ; :: thesis: for X being Subset of [:S,T:] holds proj1 (uparrow X) = uparrow (proj1 X)
let X be Subset of [:S,T:]; :: thesis: proj1 (uparrow X) = uparrow (proj1 X)
thus proj1 (uparrow X) c= uparrow (proj1 X) by Th33; :: according to XBOOLE_0:def 10 :: thesis: uparrow (proj1 X) c= proj1 (uparrow X)
let a be object ; :: according to TARSKI:def 3 :: thesis: ( not a in uparrow (proj1 X) or a in proj1 (uparrow X) )
assume A1: a in uparrow (proj1 X) ; :: thesis: a in proj1 (uparrow X)
then reconsider S9 = S as non empty RelStr ;
reconsider a9 = a as Element of S9 by A1;
consider b being Element of S9 such that
A2: b <= a9 and
A3: b in proj1 X by A1, WAYBEL_0:def 16;
consider b2 being object such that
A4: [b,b2] in X by A3, XTUPLE_0:def 12;
A5: the carrier of [:S,T:] = [: the carrier of S, the carrier of T:] by YELLOW_3:def 2;
then reconsider T9 = T as non empty reflexive RelStr by A4, ZFMISC_1:87;
reconsider b2 = b2 as Element of T9 by A5, A4, ZFMISC_1:87;
b2 <= b2 ;
then [b,b2] <= [a9,b2] by A2, YELLOW_3:11;
then [a9,b2] in uparrow X by A4, WAYBEL_0:def 16;
hence a in proj1 (uparrow X) by XTUPLE_0:def 12; :: thesis: verum