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 (Ex (a,PA,G)) 'imp' (All (b,PA,G)) '<' All ((a 'imp' 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 (Ex (a,PA,G)) 'imp' (All (b,PA,G)) '<' All ((a 'imp' b),PA,G)
let a, b be Function of Y,BOOLEAN; :: thesis: for PA being a_partition of Y holds (Ex (a,PA,G)) 'imp' (All (b,PA,G)) '<' All ((a 'imp' b),PA,G)
let PA be a_partition of Y; :: thesis: (Ex (a,PA,G)) 'imp' (All (b,PA,G)) '<' All ((a 'imp' b),PA,G)
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not ((Ex (a,PA,G)) 'imp' (All (b,PA,G))) . z = TRUE or (All ((a 'imp' b),PA,G)) . z = TRUE )
assume ((Ex (a,PA,G)) 'imp' (All (b,PA,G))) . z = TRUE ; :: thesis: (All ((a 'imp' b),PA,G)) . z = TRUE
then A1: ('not' ((Ex (a,PA,G)) . z)) 'or' ((All (b,PA,G)) . z) = TRUE by BVFUNC_1:def 8;

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

suppose A2: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
b . x = TRUE ; :: thesis: (All ((a 'imp' b),PA,G)) . z = TRUE

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

proof

let x be Element of Y; :: thesis: ( x in EqClass (z,(CompF (PA,G))) implies (a 'imp' b) . x = TRUE )
assume A3: x in EqClass (z,(CompF (PA,G))) ; :: thesis: (a 'imp' b) . x = TRUE
thus (a 'imp' b) . x = ('not' (a . x)) 'or' (b . x) by BVFUNC_1:def 8
.= ('not' (a . x)) 'or' TRUE by A2, A3
.= TRUE by BINARITH:10 ; :: thesis: verum

end;

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

end;

suppose A4: ( ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & a . x = TRUE ) & ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not b . x = TRUE ) ) ; :: thesis: (All ((a 'imp' 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;
then A5: 'not' ((Ex (a,PA,G)) . z) = FALSE by MARGREL1:11;
(B_INF (b,(CompF (PA,G)))) . z = FALSE by A4, BVFUNC_1:def 16;
then (All (b,PA,G)) . z = FALSE by BVFUNC_2:def 9;
hence (All ((a 'imp' b),PA,G)) . z = TRUE by A1, A5; :: thesis: verum

end;

suppose A6: ( ( for x being Element of Y holds
( not x in EqClass (z,(CompF (PA,G))) or 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 'imp' b),PA,G)) . z = TRUE

now :: thesis: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
(a 'imp' b) . x = TRUE

let x be Element of Y; :: thesis: ( x in EqClass (z,(CompF (PA,G))) implies (a 'imp' b) . x = TRUE )
assume x in EqClass (z,(CompF (PA,G))) ; :: thesis: (a 'imp' b) . x = TRUE
then A7: a . x <> TRUE by A6;
thus (a 'imp' b) . x = ('not' (a . x)) 'or' (b . x) by BVFUNC_1:def 8
.= ('not' FALSE) 'or' (b . x) by A7, XBOOLEAN:def 3
.= TRUE 'or' (b . x) by MARGREL1:11
.= TRUE by BINARITH:10 ; :: thesis: verum

end;

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

end;

end;