let Y be non empty set ; :: thesis:
let G be Subset of (PARTITIONS Y); :: thesis:
let u, a be Element of Funcs Y,BOOLEAN ; :: thesis:
let PA be a_partition of Y; :: thesis:
assume u is_independent_of PA,G ; :: thesis:
then A1: u is_dependent_of CompF PA,G by Def8;
A2: for z being Element of Y holds (B_INF (u 'or' a),(CompF PA,G)) . z = (u 'or' (B_INF a,(CompF PA,G))) . z

let z be Element of Y; :: thesis:
A3: (u 'or' (B_INF a,(CompF PA,G))) . z = (u . z) 'or' ((B_INF a,(CompF PA,G)) . z) by BVFUNC_1:def 7;

per cases ;

suppose A4: for x being Element of Y st x in EqClass z,(CompF PA,G) holds
a . x = TRUE ; :: thesis:

then (B_INF a,(CompF PA,G)) . z = TRUE by BVFUNC_1:def 19;
then A5: (u 'or' (B_INF a,(CompF PA,G))) . z = TRUE by A3, BINARITH:19;
for x being Element of Y st x in EqClass z,(CompF PA,G) holds
(u 'or' a) . x = TRUE

let x be Element of Y; :: thesis:
assume A6: x in EqClass z,(CompF PA,G) ; :: thesis:
(u 'or' a) . x = (u . x) 'or' (a . x) by BVFUNC_1:def 7
.= (u . x) 'or' TRUE by A4, A6
.= TRUE by BINARITH:19 ;
hence (u 'or' a) . x = TRUE ; :: thesis:

end;

hence (B_INF (u 'or' a),(CompF PA,G)) . z = (u 'or' (B_INF a,(CompF PA,G))) . z by A5, BVFUNC_1:def 19; :: thesis:

end;

suppose A7: ( 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
u . x = TRUE ) ) ; :: thesis:

z in EqClass z,(CompF PA,G) by EQREL_1:def 8;
then (u 'or' (B_INF a,(CompF PA,G))) . z = TRUE 'or' ((B_INF a,(CompF PA,G)) . z) by A3, A7;
then A8: (u 'or' (B_INF a,(CompF PA,G))) . z = TRUE by BINARITH:19;
for x being Element of Y st x in EqClass z,(CompF PA,G) holds
(u 'or' a) . x = TRUE

let x be Element of Y; :: thesis:
assume A9: x in EqClass z,(CompF PA,G) ; :: thesis:
(u 'or' a) . x = (u . x) 'or' (a . x) by BVFUNC_1:def 7
.= TRUE 'or' (a . x) by A7, A9
.= TRUE by BINARITH:19 ;
hence (u 'or' a) . x = TRUE ; :: thesis:

end;

hence (B_INF (u 'or' a),(CompF PA,G)) . z = (u 'or' (B_INF a,(CompF PA,G))) . z by A8, BVFUNC_1:def 19; :: thesis:

end;

suppose A10: ( 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 u . x = TRUE ) ) ; :: thesis:

then A11: (B_INF a,(CompF PA,G)) . z = FALSE by BVFUNC_1:def 19;
consider x1 being Element of Y such that
A12: ( x1 in EqClass z,(CompF PA,G) & a . x1 <> TRUE ) by A10;
consider x2 being Element of Y such that
A13: ( x2 in EqClass z,(CompF PA,G) & u . x2 <> TRUE ) by A10;
A14: ( z in EqClass z,(CompF PA,G) & EqClass z,(CompF PA,G) in CompF PA,G ) by EQREL_1:def 8;
u . x1 = u . x2 by A1, A12, A13, BVFUNC_1:def 18;
then A15: u . x1 = FALSE by A13, XBOOLEAN:def 3;
a . x1 = FALSE by A12, XBOOLEAN:def 3;
then A16: (u 'or' a) . x1 = FALSE 'or' FALSE by A15, BVFUNC_1:def 7;
u . x1 = u . z by A1, A12, A14, BVFUNC_1:def 18;
hence (B_INF (u 'or' a),(CompF PA,G)) . z = (u 'or' (B_INF a,(CompF PA,G))) . z by A3, A11, A12, A15, A16, BVFUNC_1:def 19; :: thesis:

end;

consider k3 being Function such that
A17: ( All (u 'or' a),PA,G = k3 & dom k3 = Y & rng k3 c= BOOLEAN ) by FUNCT_2:def 2;
consider k4 being Function such that
A18: ( u 'or' (All a,PA,G) = k4 & dom k4 = Y & rng k4 c= BOOLEAN ) by FUNCT_2:def 2;
for u being set st u in Y holds
k3 . u = k4 . u by A2, A17, A18;
hence All (u 'or' a),PA,G = u 'or' (All a,PA,G) by A17, A18, FUNCT_1:9; :: thesis: