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

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

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

then a . z = FALSE by MARGREL1:11;
then (B_INF (a,(CompF (PA,G)))) . z = FALSE by A2, BVFUNC_1:def 16;
then (All (a,PA,G)) . z = FALSE by BVFUNC_2:def 9;
hence ((All (a,PA,G)) 'imp' b) . z = ('not' FALSE) 'or' (b . z) by BVFUNC_1:def 8
.= TRUE 'or' (b . z) by MARGREL1:11
.= TRUE by BINARITH:10 ;
:: thesis: verum

end;

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

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

end;

end;