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 All (a,PA,G) '<' (Ex (b,PA,G)) 'imp' (Ex ((a '&' 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 All (a,PA,G) '<' (Ex (b,PA,G)) 'imp' (Ex ((a '&' b),PA,G))

let a, b be Function of Y,BOOLEAN; :: thesis: for PA being a_partition of Y holds All (a,PA,G) '<' (Ex (b,PA,G)) 'imp' (Ex ((a '&' b),PA,G))

let PA be a_partition of Y; :: thesis: All (a,PA,G) '<' (Ex (b,PA,G)) 'imp' (Ex ((a '&' b),PA,G))

let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not (All (a,PA,G)) . z = TRUE or ((Ex (b,PA,G)) 'imp' (Ex ((a '&' b),PA,G))) . z = TRUE )

assume A1: (All (a,PA,G)) . z = TRUE ; :: thesis: ((Ex (b,PA,G)) 'imp' (Ex ((a '&' b),PA,G))) . z = TRUE

for a, b being Function of Y,BOOLEAN

for PA being a_partition of Y holds All (a,PA,G) '<' (Ex (b,PA,G)) 'imp' (Ex ((a '&' 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 All (a,PA,G) '<' (Ex (b,PA,G)) 'imp' (Ex ((a '&' b),PA,G))

let a, b be Function of Y,BOOLEAN; :: thesis: for PA being a_partition of Y holds All (a,PA,G) '<' (Ex (b,PA,G)) 'imp' (Ex ((a '&' b),PA,G))

let PA be a_partition of Y; :: thesis: All (a,PA,G) '<' (Ex (b,PA,G)) 'imp' (Ex ((a '&' b),PA,G))

let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not (All (a,PA,G)) . z = TRUE or ((Ex (b,PA,G)) 'imp' (Ex ((a '&' b),PA,G))) . z = TRUE )

assume A1: (All (a,PA,G)) . z = TRUE ; :: thesis: ((Ex (b,PA,G)) 'imp' (Ex ((a '&' b),PA,G))) . z = TRUE

A2: now :: thesis: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds

a . x = TRUE

a . x = TRUE

assume
ex x being Element of Y st

( x in EqClass (z,(CompF (PA,G))) & not a . x = TRUE ) ; :: thesis: contradiction

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

hence contradiction by A1, BVFUNC_2:def 9; :: thesis: verum

end;( x in EqClass (z,(CompF (PA,G))) & not a . x = TRUE ) ; :: thesis: contradiction

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

hence contradiction by A1, BVFUNC_2:def 9; :: thesis: verum

per cases
( (Ex (b,PA,G)) . z = TRUE or (Ex (b,PA,G)) . z <> TRUE )
;

end;

suppose A3:
(Ex (b,PA,G)) . z = TRUE
; :: thesis: ((Ex (b,PA,G)) 'imp' (Ex ((a '&' b),PA,G))) . z = TRUE

A4: x1 in EqClass (z,(CompF (PA,G))) and

A5: b . x1 = TRUE ;

(a '&' b) . x1 = (a . x1) '&' (b . x1) by MARGREL1:def 20

.= TRUE '&' TRUE by A2, A4, A5

.= TRUE ;

then (B_SUP ((a '&' b),(CompF (PA,G)))) . z = TRUE by A4, BVFUNC_1:def 17;

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

hence ((Ex (b,PA,G)) 'imp' (Ex ((a '&' b),PA,G))) . z = ('not' ((Ex (b,PA,G)) . z)) 'or' TRUE by BVFUNC_1:def 8

.= TRUE by BINARITH:10 ;

:: thesis: verum

end;

now :: thesis: ex x being Element of Y st

( x in EqClass (z,(CompF (PA,G))) & b . x = TRUE )

then consider x1 being Element of Y such that ( x in EqClass (z,(CompF (PA,G))) & b . x = TRUE )

assume
for x being Element of Y holds

( not x in EqClass (z,(CompF (PA,G))) or not b . x = TRUE ) ; :: thesis: contradiction

then (B_SUP (b,(CompF (PA,G)))) . z = FALSE by BVFUNC_1:def 17;

hence contradiction by A3, BVFUNC_2:def 10; :: thesis: verum

end;( not x in EqClass (z,(CompF (PA,G))) or not b . x = TRUE ) ; :: thesis: contradiction

then (B_SUP (b,(CompF (PA,G)))) . z = FALSE by BVFUNC_1:def 17;

hence contradiction by A3, BVFUNC_2:def 10; :: thesis: verum

A4: x1 in EqClass (z,(CompF (PA,G))) and

A5: b . x1 = TRUE ;

(a '&' b) . x1 = (a . x1) '&' (b . x1) by MARGREL1:def 20

.= TRUE '&' TRUE by A2, A4, A5

.= TRUE ;

then (B_SUP ((a '&' b),(CompF (PA,G)))) . z = TRUE by A4, BVFUNC_1:def 17;

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

hence ((Ex (b,PA,G)) 'imp' (Ex ((a '&' b),PA,G))) . z = ('not' ((Ex (b,PA,G)) . z)) 'or' TRUE by BVFUNC_1:def 8

.= TRUE by BINARITH:10 ;

:: thesis: verum

suppose
(Ex (b,PA,G)) . z <> TRUE
; :: thesis: ((Ex (b,PA,G)) 'imp' (Ex ((a '&' b),PA,G))) . z = TRUE

then
(Ex (b,PA,G)) . z = FALSE
by XBOOLEAN:def 3;

hence ((Ex (b,PA,G)) 'imp' (Ex ((a '&' b),PA,G))) . z = ('not' FALSE) 'or' ((Ex ((a '&' b),PA,G)) . z) by BVFUNC_1:def 8

.= TRUE 'or' ((Ex ((a '&' b),PA,G)) . z) by MARGREL1:11

.= TRUE by BINARITH:10 ;

:: thesis: verum

end;hence ((Ex (b,PA,G)) 'imp' (Ex ((a '&' b),PA,G))) . z = ('not' FALSE) 'or' ((Ex ((a '&' b),PA,G)) . z) by BVFUNC_1:def 8

.= TRUE 'or' ((Ex ((a '&' b),PA,G)) . z) by MARGREL1:11

.= TRUE by BINARITH:10 ;

:: thesis: verum