let Y be non empty set ; :: thesis: for a being Function of Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y holds 'not' (Ex ((All (a,A,G)),B,G)) '<' Ex ((Ex (('not' a),B,G)),A,G)
let a be Function of Y,BOOLEAN; :: thesis: for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y holds 'not' (Ex ((All (a,A,G)),B,G)) '<' Ex ((Ex (('not' a),B,G)),A,G)
let G be Subset of (PARTITIONS Y); :: thesis: for A, B being a_partition of Y holds 'not' (Ex ((All (a,A,G)),B,G)) '<' Ex ((Ex (('not' a),B,G)),A,G)
let A, B be a_partition of Y; :: thesis: 'not' (Ex ((All (a,A,G)),B,G)) '<' Ex ((Ex (('not' a),B,G)),A,G)
A1: All (a,A,G) = B_INF (a,(CompF (A,G))) by BVFUNC_2:def 9;
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not ('not' (Ex ((All (a,A,G)),B,G))) . z = TRUE or (Ex ((Ex (('not' a),B,G)),A,G)) . z = TRUE )
A2: z in EqClass (z,(CompF (B,G))) by EQREL_1:def 6;
assume ('not' (Ex ((All (a,A,G)),B,G))) . z = TRUE ; :: thesis: (Ex ((Ex (('not' a),B,G)),A,G)) . z = TRUE
then A3: 'not' ((Ex ((All (a,A,G)),B,G)) . z) = TRUE by MARGREL1:def 19;
Ex ((All (a,A,G)),B,G) = B_SUP ((All (a,A,G)),(CompF (B,G))) by BVFUNC_2:def 10;
then (All (a,A,G)) . z <> TRUE by A3, A2, BVFUNC_1:def 17;
then consider x1 being Element of Y such that
A4: x1 in EqClass (z,(CompF (A,G))) and
A5: a . x1 <> TRUE by A1, BVFUNC_1:def 16;
a . x1 = FALSE by A5, XBOOLEAN:def 3;
then A6: ('not' a) . x1 = 'not' FALSE by MARGREL1:def 19;
A7: Ex (('not' a),B,G) = B_SUP (('not' a),(CompF (B,G))) by BVFUNC_2:def 10;
x1 in EqClass (x1,(CompF (B,G))) by EQREL_1:def 6;
then (Ex (('not' a),B,G)) . x1 = TRUE by A7, A6, BVFUNC_1:def 17;
then (B_SUP ((Ex (('not' a),B,G)),(CompF (A,G)))) . z = TRUE by A4, BVFUNC_1:def 17;
hence (Ex ((Ex (('not' a),B,G)),A,G)) . z = TRUE by BVFUNC_2:def 10; :: thesis: verum