let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for a, b being Function of Y,BOOLEAN
for PA being a_partition of Y holds (Ex (a,PA,G)) 'xor' (Ex (b,PA,G)) '<' Ex ((a 'xor' b),PA,G)
let G be Subset of (PARTITIONS Y); :: thesis: for a, b being Function of Y,BOOLEAN
for PA being a_partition of Y holds (Ex (a,PA,G)) 'xor' (Ex (b,PA,G)) '<' Ex ((a 'xor' b),PA,G)
let a, b be Function of Y,BOOLEAN; :: thesis: for PA being a_partition of Y holds (Ex (a,PA,G)) 'xor' (Ex (b,PA,G)) '<' Ex ((a 'xor' b),PA,G)
let PA be a_partition of Y; :: thesis: (Ex (a,PA,G)) 'xor' (Ex (b,PA,G)) '<' Ex ((a 'xor' b),PA,G)
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not ((Ex (a,PA,G)) 'xor' (Ex (b,PA,G))) . z = TRUE or (Ex ((a 'xor' b),PA,G)) . z = TRUE )
A1: ((Ex (a,PA,G)) 'xor' (Ex (b,PA,G))) . z = ((Ex (a,PA,G)) . z) 'xor' ((Ex (b,PA,G)) . z) by BVFUNC_1:def 5
.= (('not' ((Ex (a,PA,G)) . z)) '&' ((Ex (b,PA,G)) . z)) 'or' (((Ex (a,PA,G)) . z) '&' ('not' ((Ex (b,PA,G)) . z))) ;
A2: ( ('not' ((Ex (a,PA,G)) . z)) '&' ((Ex (b,PA,G)) . z) = TRUE or ('not' ((Ex (a,PA,G)) . z)) '&' ((Ex (b,PA,G)) . z) = FALSE ) by XBOOLEAN:def 3;
A3: 'not' FALSE = TRUE by MARGREL1:11;
assume A4: ((Ex (a,PA,G)) 'xor' (Ex (b,PA,G))) . z = TRUE ; :: thesis: (Ex ((a 'xor' b),PA,G)) . z = TRUE

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

per cases ( ('not' ((Ex (a,PA,G)) . z)) '&' ((Ex (b,PA,G)) . z) = TRUE or ((Ex (a,PA,G)) . z) '&' ('not' ((Ex (b,PA,G)) . z)) = TRUE ) by A4, A1, A2, BINARITH:3;

case A5: ('not' ((Ex (a,PA,G)) . z)) '&' ((Ex (b,PA,G)) . z) = TRUE ; :: thesis: (Ex ((a 'xor' b),PA,G)) . z = TRUE

then (Ex (b,PA,G)) . z = TRUE by MARGREL1:12;
then consider x1 being Element of Y such that
A6: x1 in EqClass (z,(CompF (PA,G))) and
A7: b . x1 = TRUE by BVFUNC_1:def 17;
'not' ((Ex (a,PA,G)) . z) = TRUE by A5, MARGREL1:12;
then a . x1 <> TRUE by A6, BVFUNC_1:def 17, MARGREL1:11;
then A8: a . x1 = FALSE by XBOOLEAN:def 3;
(a 'xor' b) . x1 = (a . x1) 'xor' (b . x1) by BVFUNC_1:def 5
.= TRUE 'or' FALSE by A3, A7, A8
.= TRUE by BINARITH:10 ;
hence (Ex ((a 'xor' b),PA,G)) . z = TRUE by A6, BVFUNC_1:def 17; :: thesis: verum

end;

case A9: ((Ex (a,PA,G)) . z) '&' ('not' ((Ex (b,PA,G)) . z)) = TRUE ; :: thesis: (Ex ((a 'xor' b),PA,G)) . z = TRUE

then (Ex (a,PA,G)) . z = TRUE by MARGREL1:12;
then consider x1 being Element of Y such that
A10: x1 in EqClass (z,(CompF (PA,G))) and
A11: a . x1 = TRUE by BVFUNC_1:def 17;
'not' ((Ex (b,PA,G)) . z) = TRUE by A9, MARGREL1:12;
then b . x1 <> TRUE by A10, BVFUNC_1:def 17, MARGREL1:11;
then A12: b . x1 = FALSE by XBOOLEAN:def 3;
(a 'xor' b) . x1 = (a . x1) 'xor' (b . x1) by BVFUNC_1:def 5
.= FALSE 'or' TRUE by A3, A11, A12
.= TRUE by BINARITH:10 ;
hence (Ex ((a 'xor' b),PA,G)) . z = TRUE by A10, BVFUNC_1:def 17; :: thesis: verum

end;

end;

end;

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