let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for a, u being Function of 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 a, u being Function of 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 a, u be Function of 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 MARGREL1:11;
assume A2: u is_independent_of PA,G ; :: thesis: u 'xor' (Ex (a,PA,G)) '<' Ex ((u 'xor' a),PA,G)
let z be Element of Y; :: according to BVFUNC_1:def 12 :: 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 5
.= (('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 6;
assume A6: (u 'xor' (Ex (a,PA,G))) . z = TRUE ; :: thesis: (Ex ((u 'xor' a),PA,G)) . z = TRUE

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

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:3;

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 MARGREL1:12;
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 17;
A10: u . z = u . x1 by A2, A5, A8, BVFUNC_1:def 15;
A11: 'not' (u . z) = TRUE by A7, MARGREL1:12;
(u 'xor' a) . x1 = (u . x1) 'xor' (a . x1) by BVFUNC_1:def 5
.= TRUE 'or' FALSE by A11, A9, A10, MARGREL1:11
.= TRUE by BINARITH:10 ;
hence (Ex ((u 'xor' a),PA,G)) . z = TRUE by A8, BVFUNC_1:def 17; :: 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 MARGREL1:12;
then a . z <> TRUE by A5, BVFUNC_1:def 17, MARGREL1:11;
then A13: a . z = FALSE by XBOOLEAN:def 3;
A14: u . z = TRUE by A12, MARGREL1:12;
(u 'xor' a) . z = (u . z) 'xor' (a . z) by BVFUNC_1:def 5
.= FALSE 'or' TRUE by A1, A14, A13
.= TRUE by BINARITH:10 ;
hence (Ex ((u 'xor' a),PA,G)) . z = TRUE by A5, BVFUNC_1:def 17; :: thesis: verum

end;

end;

end;

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