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,PA,G)) 'imp' (All (b,PA,G)) '<' (All (a,PA,G)) 'imp' (Ex (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,PA,G)) 'imp' (All (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,PA,G)) 'imp' (All (b,PA,G)) '<' (All (a,PA,G)) 'imp' (Ex (b,PA,G))
let PA be a_partition of Y; :: thesis: (All (a,PA,G)) 'imp' (All (b,PA,G)) '<' (All (a,PA,G)) 'imp' (Ex (b,PA,G))
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not ((All (a,PA,G)) 'imp' (All (b,PA,G))) . z = TRUE or ((All (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE )
A1: ( 'not' ((All (a,PA,G)) . z) = TRUE or 'not' ((All (a,PA,G)) . z) = FALSE ) by XBOOLEAN:def 3;
A2: z in EqClass (z,(CompF (PA,G))) by EQREL_1:def 6;
assume ((All (a,PA,G)) 'imp' (All (b,PA,G))) . z = TRUE ; :: thesis: ((All (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE
then A3: ('not' ((All (a,PA,G)) . z)) 'or' ((All (b,PA,G)) . z) = TRUE by BVFUNC_1:def 8;

per cases ( 'not' ((All (a,PA,G)) . z) = TRUE or (All (b,PA,G)) . z = TRUE ) by A3, A1, BINARITH:3;

suppose 'not' ((All (a,PA,G)) . z) = TRUE ; :: thesis: ((All (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE

hence ((All (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE 'or' ((Ex (b,PA,G)) . z) by BVFUNC_1:def 8
.= TRUE by BINARITH:10 ;
:: thesis: verum

end;

suppose A4: (All (b,PA,G)) . z = TRUE ; :: thesis: ((All (a,PA,G)) 'imp' (Ex (b,PA,G))) . z = TRUE

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

assume ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not b . x = TRUE ) ; :: thesis: contradiction
then (B_INF (b,(CompF (PA,G)))) . z = FALSE by BVFUNC_1:def 16;
hence contradiction by A4, BVFUNC_2:def 9; :: thesis: verum

end;

then b . z = TRUE by A2;
then (B_SUP (b,(CompF (PA,G)))) . z = TRUE by A2, 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;

end;