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) '<' (Ex (a,PA,G)) 'or' (All (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) '<' (Ex (a,PA,G)) 'or' (All (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) '<' (Ex (a,PA,G)) 'or' (All (b,PA,G))
let PA be a_partition of Y; :: thesis: All ((a 'or' b),PA,G) '<' (Ex (a,PA,G)) 'or' (All (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 ((Ex (a,PA,G)) 'or' (All (b,PA,G))) . z = TRUE )
assume A1: (All ((a 'or' b),PA,G)) . z = TRUE ; :: thesis: ((Ex (a,PA,G)) 'or' (All (b,PA,G))) . z = TRUE

per cases ( ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & a . x = TRUE ) or ( ( for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
b . x = TRUE ) & ( for x being Element of Y holds
( not x in EqClass (z,(CompF (PA,G))) or not a . x = TRUE ) ) ) or ( ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not b . x = TRUE ) & ( for x being Element of Y holds
( not x in EqClass (z,(CompF (PA,G))) or not a . x = TRUE ) ) ) ) ;

suppose ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & a . x = TRUE ) ; :: thesis: ((Ex (a,PA,G)) 'or' (All (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;
hence ((Ex (a,PA,G)) 'or' (All (b,PA,G))) . z = TRUE 'or' ((All (b,PA,G)) . z) 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
b . x = TRUE ) & ( for x being Element of Y holds
( not x in EqClass (z,(CompF (PA,G))) or not a . x = TRUE ) ) ) ; :: thesis: ((Ex (a,PA,G)) 'or' (All (b,PA,G))) . z = TRUE

then (B_INF (b,(CompF (PA,G)))) . z = TRUE by BVFUNC_1:def 16;
then (All (b,PA,G)) . z = TRUE by BVFUNC_2:def 9;
hence ((Ex (a,PA,G)) 'or' (All (b,PA,G))) . z = ((Ex (a,PA,G)) . z) 'or' TRUE by BVFUNC_1:def 4
.= TRUE by BINARITH:10 ;
:: thesis: verum

end;

suppose A2: ( ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not b . x = TRUE ) & ( for x being Element of Y holds
( not x in EqClass (z,(CompF (PA,G))) or not a . x = TRUE ) ) ) ; :: thesis: ((Ex (a,PA,G)) 'or' (All (b,PA,G))) . z = TRUE

then consider x1 being Element of Y such that
A3: x1 in EqClass (z,(CompF (PA,G))) and
A4: b . x1 <> TRUE ;
A5: a . x1 <> TRUE by A2, A3;
A6: b . x1 = FALSE by A4, XBOOLEAN:def 3;
(a 'or' b) . x1 = (a . x1) 'or' (b . x1) by BVFUNC_1:def 4
.= FALSE 'or' FALSE by A5, A6, XBOOLEAN:def 3
.= FALSE ;
hence ((Ex (a,PA,G)) 'or' (All (b,PA,G))) . z = TRUE by A1, A3, Lm1; :: thesis: verum

end;

end;