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) '<' (All (a,PA,G)) 'imp' (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 'imp' b),PA,G) '<' (All (a,PA,G)) 'imp' (All (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' (All (b,PA,G))
let PA be a_partition of Y; :: thesis: All ((a 'imp' b),PA,G) '<' (All (a,PA,G)) 'imp' (All (b,PA,G))
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' (All (b,PA,G))) . z = TRUE )
assume A1: (All ((a 'imp' b),PA,G)) . z = TRUE ; :: thesis: ((All (a,PA,G)) 'imp' (All (b,PA,G))) . z = TRUE
A2: ((All (a,PA,G)) 'imp' (All (b,PA,G))) . z = ('not' ((All (a,PA,G)) . z)) 'or' ((All (b,PA,G)) . z) by BVFUNC_1:def 8;

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

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

then (B_INF (b,(CompF (PA,G)))) . z = TRUE by BVFUNC_1:def 16;
then ((All (a,PA,G)) 'imp' (All (b,PA,G))) . z = ('not' ((All (a,PA,G)) . z)) 'or' TRUE by BVFUNC_1:def 8
.= TRUE by BINARITH:10 ;
hence ((All (a,PA,G)) 'imp' (All (b,PA,G))) . z = TRUE ; :: thesis: verum

end;

suppose A3: ( ( for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
a . x = TRUE ) & ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not b . x = TRUE ) ) ; :: thesis: ((All (a,PA,G)) 'imp' (All (b,PA,G))) . z = TRUE

then consider x1 being Element of Y such that
A4: x1 in EqClass (z,(CompF (PA,G))) and
A5: b . x1 <> TRUE ;
A6: a . 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 ((All (a,PA,G)) 'imp' (All (b,PA,G))) . z = TRUE by A1, A4, BVFUNC_1:def 16; :: thesis: verum

end;

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

then ((All (a,PA,G)) 'imp' (All (b,PA,G))) . z = ('not' ((All (a,PA,G)) . z)) 'or' TRUE by A2, BVFUNC_1:def 16
.= TRUE by BINARITH:10 ;
hence ((All (a,PA,G)) 'imp' (All (b,PA,G))) . z = TRUE ; :: thesis: verum

end;

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

then ((All (a,PA,G)) 'imp' (All (b,PA,G))) . z = TRUE 'or' ((All (b,PA,G)) . z) by A2, BVFUNC_1:def 16, MARGREL1:11
.= TRUE by BINARITH:10 ;
hence ((All (a,PA,G)) 'imp' (All (b,PA,G))) . z = TRUE ; :: thesis: verum

end;

end;