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
u 'xor' (Ex a,PA,G) '<' Ex (u 'xor' a),PA,G
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
u 'xor' (Ex a,PA,G) '<' Ex (u 'xor' a),PA,G
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
u 'xor' (Ex a,PA,G) '<' Ex (u 'xor' a),PA,G
let PA be a_partition of Y; :: thesis: ( u is_independent_of PA,G implies u 'xor' (Ex a,PA,G) '<' Ex (u 'xor' a),PA,G )
assume u is_independent_of PA,G ; :: thesis: u 'xor' (Ex a,PA,G) '<' Ex (u 'xor' a),PA,G
then A1: u is_dependent_of CompF PA,G by Def8;
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not (u 'xor' (Ex a,PA,G)) . z = TRUE or (Ex (u 'xor' a),PA,G) . z = TRUE )
assume A2: (u 'xor' (Ex a,PA,G)) . z = TRUE ; :: thesis: (Ex (u 'xor' a),PA,G) . z = TRUE
A3: (u 'xor' (Ex a,PA,G)) . z = (u . z) 'xor' ((Ex a,PA,G) . z) by BVFUNC_1:def 8
.= (('not' (u . z)) '&' ((Ex a,PA,G) . z)) 'or' ((u . z) '&' ('not' ((Ex a,PA,G) . z))) ;
A4: ( (u . z) '&' ('not' ((Ex a,PA,G) . z)) = TRUE or (u . z) '&' ('not' ((Ex a,PA,G) . z)) = FALSE ) by XBOOLEAN:def 3;
A5: ( z in EqClass z,(CompF PA,G) & EqClass z,(CompF PA,G) in CompF PA,G ) by EQREL_1:def 8;
A6: ( 'not' FALSE = TRUE & 'not' TRUE = FALSE ) by MARGREL1:41;

now

per cases ( ('not' (u . z)) '&' ((Ex a,PA,G) . z) = TRUE or (u . z) '&' ('not' ((Ex a,PA,G) . z)) = TRUE ) by A2, A3, A4, BINARITH:7;

case ('not' (u . z)) '&' ((Ex a,PA,G) . z) = TRUE ; :: thesis: (Ex (u 'xor' a),PA,G) . z = TRUE

then A7: ( 'not' (u . z) = TRUE & (Ex a,PA,G) . z = TRUE ) by MARGREL1:45;
then consider x1 being Element of Y such that
A8: ( x1 in EqClass z,(CompF PA,G) & a . x1 = TRUE ) by BVFUNC_1:def 20;
A9: u . z = u . x1 by A1, A5, A8, BVFUNC_1:def 18;
(u 'xor' a) . x1 = (u . x1) 'xor' (a . x1) by BVFUNC_1:def 8
.= TRUE 'or' FALSE by A7, A8, A9, MARGREL1:41
.= TRUE by BINARITH:19 ;
hence (Ex (u 'xor' a),PA,G) . z = TRUE by A8, BVFUNC_1:def 20; :: thesis: verum

end;

case (u . z) '&' ('not' ((Ex a,PA,G) . z)) = TRUE ; :: thesis: (Ex (u 'xor' a),PA,G) . z = TRUE

then A10: ( u . z = TRUE & 'not' ((Ex a,PA,G) . z) = TRUE ) by MARGREL1:45;
then (Ex a,PA,G) . z = FALSE by MARGREL1:41;
then a . z <> TRUE by A5, BVFUNC_1:def 20;
then A11: a . z = FALSE by XBOOLEAN:def 3;
(u 'xor' a) . z = (u . z) 'xor' (a . z) by BVFUNC_1:def 8
.= FALSE 'or' TRUE by A6, A10, A11
.= TRUE by BINARITH:19 ;
hence (Ex (u 'xor' a),PA,G) . z = TRUE by A5, BVFUNC_1:def 20; :: thesis: verum

end;

end;

end;

hence (Ex (u 'xor' a),PA,G) . z = TRUE ; :: thesis: verum