let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for u, a being Function of Y,BOOLEAN
for PA being a_partition of Y st u is_independent_of PA,G holds
All ((u 'or' a),PA,G) = u 'or' (All (a,PA,G))
let G be Subset of (PARTITIONS Y); :: thesis: for u, a being Function of Y,BOOLEAN
for PA being a_partition of Y st u is_independent_of PA,G holds
All ((u 'or' a),PA,G) = u 'or' (All (a,PA,G))
let u, a be Function of Y,BOOLEAN; :: thesis: for PA being a_partition of Y st u is_independent_of PA,G holds
All ((u 'or' a),PA,G) = u 'or' (All (a,PA,G))
let PA be a_partition of Y; :: thesis: ( u is_independent_of PA,G implies All ((u 'or' a),PA,G) = u 'or' (All (a,PA,G)) )
assume Z: u is_independent_of PA,G ; :: thesis: All ((u 'or' a),PA,G) = u 'or' (All (a,PA,G))
let z be Element of Y; :: according to FUNCT_2:def 8 :: thesis: (All ((u 'or' a),PA,G)) . z = (u 'or' (All (a,PA,G))) . z
A6: (u 'or' (B_INF (a,(CompF (PA,G))))) . z = (u . z) 'or' ((B_INF (a,(CompF (PA,G)))) . z) by BVFUNC_1:def 4;

per cases ( for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
a . x = TRUE or ( 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 ) ) or ( 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 ) ) ) ;

suppose A7: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
a . x = TRUE ; :: thesis: (All ((u 'or' a),PA,G)) . z = (u 'or' (All (a,PA,G))) . z

A8: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
(u 'or' a) . x = TRUE

proof

let x be Element of Y; :: thesis: ( x in EqClass (z,(CompF (PA,G))) implies (u 'or' a) . x = TRUE )
assume A9: x in EqClass (z,(CompF (PA,G))) ; :: thesis: (u 'or' a) . x = TRUE
(u 'or' a) . x = (u . x) 'or' (a . x) by BVFUNC_1:def 4
.= (u . x) 'or' TRUE by A7, A9
.= TRUE by BINARITH:10 ;
hence (u 'or' a) . x = TRUE ; :: thesis: verum

end;

(B_INF (a,(CompF (PA,G)))) . z = TRUE by A7, BVFUNC_1:def 16;
then (u 'or' (B_INF (a,(CompF (PA,G))))) . z = TRUE by A6, BINARITH:10;
hence (All ((u 'or' a),PA,G)) . z = (u 'or' (All (a,PA,G))) . z by A8, BVFUNC_1:def 16; :: thesis: verum

end;

suppose A10: ( 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: (All ((u 'or' a),PA,G)) . z = (u 'or' (All (a,PA,G))) . z

A11: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
(u 'or' a) . x = TRUE

proof

let x be Element of Y; :: thesis: ( x in EqClass (z,(CompF (PA,G))) implies (u 'or' a) . x = TRUE )
assume A12: x in EqClass (z,(CompF (PA,G))) ; :: thesis: (u 'or' a) . x = TRUE
(u 'or' a) . x = (u . x) 'or' (a . x) by BVFUNC_1:def 4
.= TRUE 'or' (a . x) by A10, A12
.= TRUE by BINARITH:10 ;
hence (u 'or' a) . x = TRUE ; :: thesis: verum

end;

(u 'or' (B_INF (a,(CompF (PA,G))))) . z = TRUE 'or' ((B_INF (a,(CompF (PA,G)))) . z) by A6, A10, EQREL_1:def 6;
then (u 'or' (B_INF (a,(CompF (PA,G))))) . z = TRUE by BINARITH:10;
hence (All ((u 'or' a),PA,G)) . z = (u 'or' (All (a,PA,G))) . z by A11, BVFUNC_1:def 16; :: thesis: verum

end;

suppose A13: ( 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: (All ((u 'or' a),PA,G)) . z = (u 'or' (All (a,PA,G))) . z

then consider x2 being Element of Y such that
A14: x2 in EqClass (z,(CompF (PA,G))) and
A15: u . x2 <> TRUE ;
consider x1 being Element of Y such that
A16: x1 in EqClass (z,(CompF (PA,G))) and
A17: a . x1 <> TRUE by A13;
u . x1 = u . x2 by Z, Def8, A16, A14, BVFUNC_1:def 15;
then A18: u . x1 = FALSE by A15, XBOOLEAN:def 3;
A19: (B_INF (a,(CompF (PA,G)))) . z = FALSE by A13, BVFUNC_1:def 16;
z in EqClass (z,(CompF (PA,G))) by EQREL_1:def 6;
then A20: u . x1 = u . z by Z, Def8, A16, BVFUNC_1:def 15;
a . x1 = FALSE by A17, XBOOLEAN:def 3;
then (u 'or' a) . x1 = FALSE 'or' FALSE by A18, BVFUNC_1:def 4;
hence (All ((u 'or' a),PA,G)) . z = (u 'or' (All (a,PA,G))) . z by A6, A19, A16, A18, A20, BVFUNC_1:def 16; :: thesis: verum

end;

end;