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

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

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

A1: Ex (a,PA,G) = B_SUP (a,(CompF (PA,G))) by BVFUNC_2:def 10;

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

assume A2: (All ((a 'imp' b),PA,G)) . z = TRUE ; :: thesis: ((Ex (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE

for a, b being Function of Y,BOOLEAN

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

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

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

A1: Ex (a,PA,G) = B_SUP (a,(CompF (PA,G))) by BVFUNC_2:def 10;

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

assume A2: (All ((a 'imp' b),PA,G)) . z = TRUE ; :: thesis: ((Ex (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE

per cases
( ex x being Element of Y st

( x in EqClass (z,(CompF (PA,G))) & b . x = TRUE ) or ( ex x being Element of Y st

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

( not x in EqClass (z,(CompF (PA,G))) or 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 ) ) & ( for x being Element of Y holds

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

end;

( x in EqClass (z,(CompF (PA,G))) & b . x = TRUE ) or ( ex x being Element of Y st

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

( not x in EqClass (z,(CompF (PA,G))) or 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 ) ) & ( for x being Element of Y holds

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

suppose
ex x being Element of Y st

( x in EqClass (z,(CompF (PA,G))) & b . x = TRUE ) ; :: thesis: ((Ex (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE

( x in EqClass (z,(CompF (PA,G))) & b . x = TRUE ) ; :: thesis: ((Ex (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE

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

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

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

.= TRUE by BINARITH:10 ;

:: thesis: verum

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

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

.= TRUE by BINARITH:10 ;

:: thesis: verum

suppose A3:
( ex x being Element of Y st

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

( not x in EqClass (z,(CompF (PA,G))) or not b . x = TRUE ) ) ) ; :: thesis: ((Ex (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE

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

( not x in EqClass (z,(CompF (PA,G))) or not b . x = TRUE ) ) ) ; :: thesis: ((Ex (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE

then consider x1 being Element of Y such that

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

A5: a . x1 = TRUE ;

A6: b . x1 <> TRUE by A3, A4;

(a 'imp' b) . x1 = ('not' (a . x1)) 'or' (b . x1) by BVFUNC_1:def 8

.= ('not' TRUE) 'or' FALSE by A5, A6, XBOOLEAN:def 3

.= FALSE 'or' FALSE by MARGREL1:11

.= FALSE ;

hence ((Ex (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE by A2, A4, Lm2; :: thesis: verum

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

A5: a . x1 = TRUE ;

A6: b . x1 <> TRUE by A3, A4;

(a 'imp' b) . x1 = ('not' (a . x1)) 'or' (b . x1) by BVFUNC_1:def 8

.= ('not' TRUE) 'or' FALSE by A5, A6, XBOOLEAN:def 3

.= FALSE 'or' FALSE by MARGREL1:11

.= FALSE ;

hence ((Ex (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE by A2, A4, Lm2; :: thesis: verum

suppose A7:
( ( for x being Element of Y holds

( not x in EqClass (z,(CompF (PA,G))) or not a . x = TRUE ) ) & ( for x being Element of Y holds

( not x in EqClass (z,(CompF (PA,G))) or not b . x = TRUE ) ) ) ; :: thesis: ((Ex (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE

( not x in EqClass (z,(CompF (PA,G))) or not a . x = TRUE ) ) & ( for x being Element of Y holds

( not x in EqClass (z,(CompF (PA,G))) or not b . x = TRUE ) ) ) ; :: thesis: ((Ex (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE

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

.= ('not' FALSE) 'or' ((Ex (b,PA,G)) . z) by A1, A7, BVFUNC_1:def 17

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

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

end;.= ('not' FALSE) 'or' ((Ex (b,PA,G)) . z) by A1, A7, BVFUNC_1:def 17

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

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