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) )
A1: 'not' FALSE = TRUE by MARGREL1:41;
assume u is_independent_of PA,G ; :: thesis: u 'xor' (Ex (a,PA,G)) '<' Ex ((u 'xor' a),PA,G)
then A2: 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 )
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))) by EQREL_1:def 8;
assume A6: (u 'xor' (Ex (a,PA,G))) . z = TRUE ; :: thesis: (Ex ((u 'xor' a),PA,G)) . z = TRUE

now

per cases ( ('not' (u . z)) '&' ((Ex (a,PA,G)) . z) = TRUE or (u . z) '&' ('not' ((Ex (a,PA,G)) . z)) = TRUE ) by A6, A3, A4, BINARITH:7;

case A7: ('not' (u . z)) '&' ((Ex (a,PA,G)) . z) = TRUE ; :: thesis: (Ex ((u 'xor' a),PA,G)) . z = TRUE

then (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))) and
A9: a . x1 = TRUE by BVFUNC_1:def 20;
A10: u . z = u . x1 by A2, A5, A8, BVFUNC_1:def 18;
A11: 'not' (u . z) = TRUE by A7, MARGREL1:45;
(u 'xor' a) . x1 = (u . x1) 'xor' (a . x1) by BVFUNC_1:def 8
.= TRUE 'or' FALSE by A11, A9, A10, 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 A12: (u . z) '&' ('not' ((Ex (a,PA,G)) . z)) = TRUE ; :: thesis: (Ex ((u 'xor' a),PA,G)) . z = TRUE

then '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 A13: a . z = FALSE by XBOOLEAN:def 3;
A14: u . z = TRUE by A12, MARGREL1:45;
(u 'xor' a) . z = (u . z) 'xor' (a . z) by BVFUNC_1:def 8
.= FALSE 'or' TRUE by A1, A14, A13
.= 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