let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for u, a being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y st u is_independent_of PA,G holds
Ex (a 'imp' u),PA,G '<' (All a,PA,G) 'imp' u
let G be Subset of (PARTITIONS Y); :: thesis: for u, a being Element of Funcs Y,BOOLEAN
for PA being a_partition of Y st u is_independent_of PA,G holds
Ex (a 'imp' u),PA,G '<' (All a,PA,G) 'imp' u
let u, a be Element of Funcs Y,BOOLEAN ; :: thesis: for PA being a_partition of Y st u is_independent_of PA,G holds
Ex (a 'imp' u),PA,G '<' (All a,PA,G) 'imp' u
let PA be a_partition of Y; :: thesis: ( u is_independent_of PA,G implies Ex (a 'imp' u),PA,G '<' (All a,PA,G) 'imp' u )
assume u is_independent_of PA,G ; :: thesis: Ex (a 'imp' u),PA,G '<' (All a,PA,G) 'imp' u
then A1: u is_dependent_of CompF PA,G by BVFUNC_2:def 8;
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not (Ex (a 'imp' u),PA,G) . z = TRUE or ((All a,PA,G) 'imp' u) . z = TRUE )
assume A2: (Ex (a 'imp' u),PA,G) . z = TRUE ; :: thesis: ((All a,PA,G) 'imp' u) . z = TRUE

now

assume for x being Element of Y holds
( not x in EqClass z,(CompF PA,G) or not (a 'imp' u) . x = TRUE ) ; :: thesis: contradiction
then (B_SUP (a 'imp' u),(CompF PA,G)) . z = FALSE by BVFUNC_1:def 20;
then (Ex (a 'imp' u),PA,G) . z = FALSE by BVFUNC_2:def 10;
hence contradiction by A2; :: thesis: verum

end;

then consider x1 being Element of Y such that
A3: ( x1 in EqClass z,(CompF PA,G) & (a 'imp' u) . x1 = TRUE ) ;
A4: ('not' (a . x1)) 'or' (u . x1) = TRUE by A3, BVFUNC_1:def 11;
A5: ( 'not' (a . x1) = TRUE or 'not' (a . x1) = FALSE ) by XBOOLEAN:def 3;
A6: ( z in EqClass z,(CompF PA,G) & EqClass z,(CompF PA,G) in CompF PA,G ) by EQREL_1:def 8;

per cases ( 'not' (a . x1) = TRUE or u . x1 = TRUE ) by A4, A5, BINARITH:7;

suppose 'not' (a . x1) = TRUE ; :: thesis: ((All a,PA,G) 'imp' u) . z = TRUE

then a . x1 = FALSE by MARGREL1:41;
then (B_INF a,(CompF PA,G)) . z = FALSE by A3, BVFUNC_1:def 19;
then (All a,PA,G) . z = FALSE by BVFUNC_2:def 9;
hence ((All a,PA,G) 'imp' u) . z = ('not' FALSE ) 'or' (u . z) by BVFUNC_1:def 11
.= TRUE 'or' (u . z) by MARGREL1:41
.= TRUE by BINARITH:19 ;
:: thesis: verum

end;

suppose A7: u . x1 = TRUE ; :: thesis: ((All a,PA,G) 'imp' u) . z = TRUE

u . x1 = u . z by A1, A3, A6, BVFUNC_1:def 18;
hence ((All a,PA,G) 'imp' u) . z = ('not' ((All a,PA,G) . z)) 'or' TRUE by A7, BVFUNC_1:def 11
.= TRUE by BINARITH:19 ;
:: thesis: verum

end;

end;