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 'or' b),PA,G) '<' (All (a,PA,G)) 'or' (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 'or' b),PA,G) '<' (All (a,PA,G)) 'or' (Ex (b,PA,G))

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

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

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

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

for a, b being Function of Y,BOOLEAN

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

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

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

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

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

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

(a 'or' b) . x = TRUE

(a 'or' b) . x = TRUE

assume
ex x being Element of Y st

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

then (B_INF ((a 'or' b),(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 'or' b) . x = TRUE ) ; :: thesis: contradiction

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

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

per cases
( ex x being Element of Y st

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

a . x = TRUE ) & ( for x being Element of Y holds

( not x in EqClass (z,(CompF (PA,G))) or not b . x = TRUE ) ) ) or ( ex x being Element of Y st

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

a . x = TRUE ) & ( for x being Element of Y holds

( not x in EqClass (z,(CompF (PA,G))) or not b . x = TRUE ) ) ) or ( ex x being Element of Y st

( x in EqClass (z,(CompF (PA,G))) & 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: ((All (a,PA,G)) 'or' (Ex (b,PA,G))) . z = TRUE

( x in EqClass (z,(CompF (PA,G))) & b . x = TRUE ) ; :: thesis: ((All (a,PA,G)) 'or' (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 ((All (a,PA,G)) 'or' (Ex (b,PA,G))) . z = ((All (a,PA,G)) . z) 'or' TRUE by BVFUNC_1:def 4

.= TRUE by BINARITH:10 ;

:: thesis: verum

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

hence ((All (a,PA,G)) 'or' (Ex (b,PA,G))) . z = ((All (a,PA,G)) . z) 'or' TRUE by BVFUNC_1:def 4

.= TRUE by BINARITH:10 ;

:: thesis: verum

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

a . x = TRUE ) & ( for x being Element of Y holds

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

a . x = TRUE ) & ( for x being Element of Y holds

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

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

then (All (a,PA,G)) . z = TRUE by BVFUNC_2:def 9;

hence ((All (a,PA,G)) 'or' (Ex (b,PA,G))) . z = TRUE 'or' ((Ex (b,PA,G)) . z) by BVFUNC_1:def 4

.= TRUE by BINARITH:10 ;

:: thesis: verum

end;then (All (a,PA,G)) . z = TRUE by BVFUNC_2:def 9;

hence ((All (a,PA,G)) 'or' (Ex (b,PA,G))) . z = TRUE 'or' ((Ex (b,PA,G)) . z) by BVFUNC_1:def 4

.= TRUE by BINARITH:10 ;

:: thesis: verum

suppose A3:
( ex x being Element of Y st

( x in EqClass (z,(CompF (PA,G))) & 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: ((All (a,PA,G)) 'or' (Ex (b,PA,G))) . z = TRUE

( x in EqClass (z,(CompF (PA,G))) & 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: ((All (a,PA,G)) 'or' (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;

A7: a . x1 = FALSE by A5, XBOOLEAN:def 3;

(a 'or' b) . x1 = (a . x1) 'or' (b . x1) by BVFUNC_1:def 4

.= FALSE 'or' FALSE by A6, A7, XBOOLEAN:def 3

.= FALSE ;

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

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

A5: a . x1 <> TRUE ;

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

A7: a . x1 = FALSE by A5, XBOOLEAN:def 3;

(a 'or' b) . x1 = (a . x1) 'or' (b . x1) by BVFUNC_1:def 4

.= FALSE 'or' FALSE by A6, A7, XBOOLEAN:def 3

.= FALSE ;

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