let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)

for a, b being Function of 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 Function of 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 Function of 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 12 :: 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 8;

for a, b being Function of 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 Function of 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 Function of 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 12 :: 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 8;

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 ) ) ) ;

end;

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

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

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

end;(a 'imp' b) . x = TRUE

proof

then
(B_INF ((a 'imp' b),(CompF (PA,G)))) . z = TRUE
by BVFUNC_1:def 16;
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 8

.= ('not' (a . x)) 'or' TRUE by A2, A3

.= TRUE by BINARITH:10 ; :: thesis: verum

end;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 8

.= ('not' (a . x)) 'or' TRUE by A2, A3

.= TRUE by BINARITH:10 ; :: thesis: verum

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

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

( 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 17;

then (Ex (a,PA,G)) . z = TRUE by BVFUNC_2:def 10;

then A5: 'not' ((Ex (a,PA,G)) . z) = FALSE by MARGREL1:11;

(B_INF (b,(CompF (PA,G)))) . z = FALSE by A4, BVFUNC_1:def 16;

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;then (Ex (a,PA,G)) . z = TRUE by BVFUNC_2:def 10;

then A5: 'not' ((Ex (a,PA,G)) . z) = FALSE by MARGREL1:11;

(B_INF (b,(CompF (PA,G)))) . z = FALSE by A4, BVFUNC_1:def 16;

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

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

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

end;

( 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 :: thesis: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds

(a 'imp' b) . x = TRUE

then
(B_INF ((a 'imp' b),(CompF (PA,G)))) . z = TRUE
by BVFUNC_1:def 16;(a 'imp' b) . x = TRUE

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 8

.= ('not' FALSE) 'or' (b . x) by A7, XBOOLEAN:def 3

.= TRUE 'or' (b . x) by MARGREL1:11

.= TRUE by BINARITH:10 ; :: thesis: verum

end;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 8

.= ('not' FALSE) 'or' (b . x) by A7, XBOOLEAN:def 3

.= TRUE 'or' (b . x) by MARGREL1:11

.= TRUE by BINARITH:10 ; :: thesis: verum

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