let Y be non empty set ; :: thesis: for a being Element of Funcs 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 Element of Funcs 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 15 :: 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 8;
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 20;
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 20;
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 19;
a . x1 = FALSE by A5, XBOOLEAN:def 3;
then A6: ('not' a) . x1 = 'not' FALSE by MARGREL1:def 20;
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 8;
then (Ex ('not' a),B,G) . x1 = TRUE by A7, A6, BVFUNC_1:def 20;
then (B_SUP (Ex ('not' a),B,G),(CompF A,G)) . z = TRUE by A4, BVFUNC_1:def 20;
hence (Ex (Ex ('not' a),B,G),A,G) . z = TRUE by BVFUNC_2:def 10; :: thesis: verum