let Y be non empty set ; for G being Subset of (PARTITIONS Y)
for a, b, c being Function of Y,BOOLEAN
for PA being a_partition of Y holds ((Ex (c,PA,G)) '&' (All ((b 'imp' ('not' c)),PA,G))) '&' (All ((c 'imp' a),PA,G)) '<' Ex ((a '&' ('not' b)),PA,G)
let G be Subset of (PARTITIONS Y); for a, b, c being Function of Y,BOOLEAN
for PA being a_partition of Y holds ((Ex (c,PA,G)) '&' (All ((b 'imp' ('not' c)),PA,G))) '&' (All ((c 'imp' a),PA,G)) '<' Ex ((a '&' ('not' b)),PA,G)
let a, b, c be Function of Y,BOOLEAN; for PA being a_partition of Y holds ((Ex (c,PA,G)) '&' (All ((b 'imp' ('not' c)),PA,G))) '&' (All ((c 'imp' a),PA,G)) '<' Ex ((a '&' ('not' b)),PA,G)
let PA be a_partition of Y; ((Ex (c,PA,G)) '&' (All ((b 'imp' ('not' c)),PA,G))) '&' (All ((c 'imp' a),PA,G)) '<' Ex ((a '&' ('not' b)),PA,G)
let z be Element of Y; BVFUNC_1:def 12 ( not (((Ex (c,PA,G)) '&' (All ((b 'imp' ('not' c)),PA,G))) '&' (All ((c 'imp' a),PA,G))) . z = TRUE or (Ex ((a '&' ('not' b)),PA,G)) . z = TRUE )
assume
(((Ex (c,PA,G)) '&' (All ((b 'imp' ('not' c)),PA,G))) '&' (All ((c 'imp' a),PA,G))) . z = TRUE
; (Ex ((a '&' ('not' b)),PA,G)) . z = TRUE
then A1:
(((Ex (c,PA,G)) '&' (All ((b 'imp' ('not' c)),PA,G))) . z) '&' ((All ((c 'imp' a),PA,G)) . z) = TRUE
by MARGREL1:def 20;
then
(((Ex (c,PA,G)) . z) '&' ((All ((b 'imp' ('not' c)),PA,G)) . z)) '&' ((All ((c 'imp' a),PA,G)) . z) = TRUE
by MARGREL1:def 20;
then A2:
((Ex (c,PA,G)) . z) '&' ((All ((b 'imp' ('not' c)),PA,G)) . z) = TRUE
by MARGREL1:12;
then consider x1 being Element of Y such that
A3:
x1 in EqClass (z,(CompF (PA,G)))
and
A4:
c . x1 = TRUE
;
A5:
( 'not' (c . x1) = TRUE or 'not' (c . x1) = FALSE )
by XBOOLEAN:def 3;
then
(c 'imp' a) . x1 = TRUE
by A3;
then A6:
('not' (c . x1)) 'or' (a . x1) = TRUE
by BVFUNC_1:def 8;
A7:
( 'not' (b . x1) = TRUE or 'not' (b . x1) = FALSE )
by XBOOLEAN:def 3;
then
(b 'imp' ('not' c)) . x1 = TRUE
by A3;
then A8:
('not' (b . x1)) 'or' (('not' c) . x1) = TRUE
by BVFUNC_1:def 8;