let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for a, c, b being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds (All (a 'imp' c),PA,G) '&' (All (c 'imp' b),PA,G) '<' All (a 'imp' b),PA,G
let G be Subset of (PARTITIONS Y); :: thesis: for a, c, b being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds (All (a 'imp' c),PA,G) '&' (All (c 'imp' b),PA,G) '<' All (a 'imp' b),PA,G
let a, c, b be Element of Funcs Y,BOOLEAN ; :: thesis: for PA being a_partition of Y holds (All (a 'imp' c),PA,G) '&' (All (c 'imp' b),PA,G) '<' All (a 'imp' b),PA,G
let PA be a_partition of Y; :: thesis: (All (a 'imp' c),PA,G) '&' (All (c 'imp' b),PA,G) '<' All (a 'imp' b),PA,G
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not ((All (a 'imp' c),PA,G) '&' (All (c 'imp' b),PA,G)) . z = TRUE or (All (a 'imp' b),PA,G) . z = TRUE )
assume ((All (a 'imp' c),PA,G) '&' (All (c 'imp' b),PA,G)) . z = TRUE ; :: thesis: (All (a 'imp' b),PA,G) . z = TRUE
then A1: ((All (a 'imp' c),PA,G) . z) '&' ((All (c 'imp' b),PA,G) . z) = TRUE by MARGREL1:def 21;

A2: now

assume ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & not (c 'imp' b) . x = TRUE ) ; :: thesis: contradiction
then (B_INF (c 'imp' b),(CompF PA,G)) . z = FALSE by BVFUNC_1:def 19;
then (All (c 'imp' b),PA,G) . z = FALSE by BVFUNC_2:def 9;
hence contradiction by A1, MARGREL1:45; :: thesis: verum

end;

A3: now

assume ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & not (a 'imp' c) . x = TRUE ) ; :: thesis: contradiction
then (B_INF (a 'imp' c),(CompF PA,G)) . z = FALSE by BVFUNC_1:def 19;
then (All (a 'imp' c),PA,G) . z = FALSE by BVFUNC_2:def 9;
hence contradiction by A1, MARGREL1:45; :: thesis: verum

end;

for x being Element of Y st x in EqClass z,(CompF PA,G) holds
(a 'imp' b) . x = TRUE

proof

let x be Element of Y; :: thesis: ( x in EqClass z,(CompF PA,G) implies (a 'imp' b) . x = TRUE )
A4: ( 'not' (a . x) = TRUE or 'not' (a . x) = FALSE ) by XBOOLEAN:def 3;
A5: ( 'not' (c . x) = TRUE or 'not' (c . x) = FALSE ) by XBOOLEAN:def 3;
assume A6: x in EqClass z,(CompF PA,G) ; :: thesis: (a 'imp' b) . x = TRUE
then (a 'imp' c) . x = TRUE by A3;
then A7: ('not' (a . x)) 'or' (c . x) = TRUE by BVFUNC_1:def 11;
(c 'imp' b) . x = TRUE by A2, A6;
then A8: ('not' (c . x)) 'or' (b . x) = TRUE by BVFUNC_1:def 11;

per cases ( ( 'not' (a . x) = TRUE & 'not' (c . x) = TRUE ) or ( 'not' (a . x) = TRUE & b . x = TRUE ) or ( c . x = TRUE & 'not' (c . x) = TRUE ) or ( c . x = TRUE & b . x = TRUE ) ) by A7, A4, A8, A5, BINARITH:7;

suppose ( 'not' (a . x) = TRUE & 'not' (c . x) = TRUE ) ; :: thesis: (a 'imp' b) . x = TRUE

hence (a 'imp' b) . x = TRUE 'or' (b . x) by BVFUNC_1:def 11
.= TRUE by BINARITH:19 ;
:: thesis: verum

end;

suppose ( 'not' (a . x) = TRUE & b . x = TRUE ) ; :: thesis: (a 'imp' b) . x = TRUE

hence (a 'imp' b) . x = TRUE 'or' (b . x) by BVFUNC_1:def 11
.= TRUE by BINARITH:19 ;
:: thesis: verum

end;

suppose ( c . x = TRUE & 'not' (c . x) = TRUE ) ; :: thesis: (a 'imp' b) . x = TRUE

hence (a 'imp' b) . x = TRUE by MARGREL1:41; :: thesis: verum

end;

suppose ( c . x = TRUE & b . x = TRUE ) ; :: thesis: (a 'imp' b) . x = TRUE

hence (a 'imp' b) . x = ('not' (a . x)) 'or' TRUE by BVFUNC_1:def 11
.= TRUE by BINARITH:19 ;
:: thesis: verum

end;

end;

end;

then (B_INF (a 'imp' b),(CompF PA,G)) . z = TRUE by BVFUNC_1:def 19;
hence (All (a 'imp' b),PA,G) . z = TRUE by BVFUNC_2:def 9; :: thesis: verum