let Y be non empty set ; :: thesis: for G being Subset of ()
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 (); :: 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
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;
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 ) ) ) )
;
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
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;
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
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;
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
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 ;
(a 'or' b) . x1 = (a . x1) 'or' (b . x1) by BVFUNC_1:def 4
.= FALSE 'or' FALSE by
.= FALSE ;
hence ((All (a,PA,G)) 'or' (Ex (b,PA,G))) . z = TRUE by A2, A4; :: thesis: verum
end;
end;