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 'imp' b),PA,G) '<' (All (a,PA,G)) 'imp' (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 'imp' b),PA,G) '<' (All (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) '<' (All (a,PA,G)) 'imp' (Ex (b,PA,G))
let PA be a_partition of Y; :: thesis: All ((a 'imp' b),PA,G) '<' (All (a,PA,G)) 'imp' (Ex (b,PA,G))
A1: All (a,PA,G) = B_INF (a,(CompF (PA,G))) by BVFUNC_2:def 9;
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not (All ((a 'imp' b),PA,G)) . z = TRUE or ((All (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE )
assume A2: (All ((a 'imp' b),PA,G)) . z = TRUE ; :: thesis: ((All (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE
A3: z in EqClass (z,(CompF (PA,G))) by EQREL_1:def 6;
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)) '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 ((All (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = ('not' ((All (a,PA,G)) . z)) 'or' TRUE by BVFUNC_1:def 8
.= 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)) 'imp' (Ex (b,PA,G))) . z = TRUE
then A4: ( b . z <> TRUE & a . z = TRUE ) by A3;
(a 'imp' b) . z = ('not' (a . z)) 'or' (b . z) by BVFUNC_1:def 8
.= () 'or' FALSE by
.= FALSE 'or' FALSE by MARGREL1:11
.= FALSE ;
hence ((All (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE by A2, A3, Lm2; :: thesis: verum
end;
suppose A5: ( 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)) 'imp' (Ex (b,PA,G))) . z = TRUE
thus ((All (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = ('not' ((All (a,PA,G)) . z)) 'or' ((Ex (b,PA,G)) . z) by BVFUNC_1:def 8
.= () 'or' ((Ex (b,PA,G)) . z) by
.= TRUE 'or' ((Ex (b,PA,G)) . z) by MARGREL1:11
.= TRUE by BINARITH:10 ; :: thesis: verum
end;
end;