let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for a, b being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds (Ex a,PA,G) 'imp' (All b,PA,G) '<' All (a 'imp' b),PA,G
let G be Subset of (PARTITIONS Y); :: thesis: for a, b being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y holds (Ex a,PA,G) 'imp' (All b,PA,G) '<' All (a 'imp' b),PA,G
let a, b be Element of Funcs Y,BOOLEAN ; :: thesis: for PA being a_partition of Y holds (Ex a,PA,G) 'imp' (All b,PA,G) '<' All (a 'imp' b),PA,G
let PA be a_partition of Y; :: thesis: (Ex a,PA,G) 'imp' (All b,PA,G) '<' All (a 'imp' b),PA,G
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not ((Ex a,PA,G) 'imp' (All b,PA,G)) . z = TRUE or (All (a 'imp' b),PA,G) . z = TRUE )
assume ((Ex a,PA,G) 'imp' (All b,PA,G)) . z = TRUE ; :: thesis: (All (a 'imp' b),PA,G) . z = TRUE
then A1: ('not' ((Ex a,PA,G) . z)) 'or' ((All b,PA,G) . z) = TRUE by BVFUNC_1:def 11;

per cases ( for x being Element of Y st x in EqClass z,(CompF PA,G) holds
b . x = TRUE or ( ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & a . x = TRUE ) & ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & not b . x = TRUE ) ) or ( ( for x being Element of Y holds
( not x in EqClass z,(CompF PA,G) or not a . x = TRUE ) ) & ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & not b . x = TRUE ) ) ) ;

suppose A2: for x being Element of Y st x in EqClass z,(CompF PA,G) holds
b . x = TRUE ; :: thesis: (All (a 'imp' b),PA,G) . z = TRUE

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 )
assume A3: x in EqClass z,(CompF PA,G) ; :: thesis: (a 'imp' b) . x = TRUE
thus (a 'imp' b) . x = ('not' (a . x)) 'or' (b . x) by BVFUNC_1:def 11
.= ('not' (a . x)) 'or' TRUE by A2, A3
.= TRUE by BINARITH:19 ; :: thesis: verum

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

end;

suppose A4: ( ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & a . x = TRUE ) & ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & not b . x = TRUE ) ) ; :: thesis: (All (a 'imp' b),PA,G) . z = TRUE

then (B_SUP a,(CompF PA,G)) . z = TRUE by BVFUNC_1:def 20;
then (Ex a,PA,G) . z = TRUE by BVFUNC_2:def 10;
then A5: 'not' ((Ex a,PA,G) . z) = FALSE by MARGREL1:41;
(B_INF b,(CompF PA,G)) . z = FALSE by A4, BVFUNC_1:def 19;
then (All b,PA,G) . z = FALSE by BVFUNC_2:def 9;
hence (All (a 'imp' b),PA,G) . z = TRUE by A1, A5; :: thesis: verum

end;

suppose A6: ( ( for x being Element of Y holds
( not x in EqClass z,(CompF PA,G) or not a . x = TRUE ) ) & ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & not b . x = TRUE ) ) ; :: thesis: (All (a 'imp' b),PA,G) . z = TRUE

now

let x be Element of Y; :: thesis: ( x in EqClass z,(CompF PA,G) implies (a 'imp' b) . x = TRUE )
assume x in EqClass z,(CompF PA,G) ; :: thesis: (a 'imp' b) . x = TRUE
then A7: a . x <> TRUE by A6;
thus (a 'imp' b) . x = ('not' (a . x)) 'or' (b . x) by BVFUNC_1:def 11
.= ('not' FALSE ) 'or' (b . x) by A7, XBOOLEAN:def 3
.= TRUE 'or' (b . x) by MARGREL1:41
.= TRUE by BINARITH:19 ; :: thesis: verum

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

end;

end;