let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for a, b being Element of Funcs 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 Element of Funcs 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 Element of Funcs 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 15 :: 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 8
.= (('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 MARGREL1:41;
assume A4: ((Ex a,PA,G) 'xor' (Ex b,PA,G)) . z = TRUE ; :: thesis: (Ex (a 'xor' b),PA,G) . z = TRUE

now

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:7;

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 MARGREL1:45;
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 20;
'not' ((Ex a,PA,G) . z) = TRUE by A5, MARGREL1:45;
then (Ex a,PA,G) . z = FALSE by MARGREL1:41;
then a . x1 <> TRUE by A6, BVFUNC_1:def 20;
then A8: a . x1 = FALSE by XBOOLEAN:def 3;
(a 'xor' b) . x1 = (a . x1) 'xor' (b . x1) by BVFUNC_1:def 8
.= TRUE 'or' FALSE by A3, A7, A8
.= TRUE by BINARITH:19 ;
hence (Ex (a 'xor' b),PA,G) . z = TRUE by A6, BVFUNC_1:def 20; :: 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 MARGREL1:45;
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 20;
'not' ((Ex b,PA,G) . z) = TRUE by A9, MARGREL1:45;
then (Ex b,PA,G) . z = FALSE by MARGREL1:41;
then b . x1 <> TRUE by A10, BVFUNC_1:def 20;
then A12: b . x1 = FALSE by XBOOLEAN:def 3;
(a 'xor' b) . x1 = (a . x1) 'xor' (b . x1) by BVFUNC_1:def 8
.= FALSE 'or' TRUE by A3, A11, A12
.= TRUE by BINARITH:19 ;
hence (Ex (a 'xor' b),PA,G) . z = TRUE by A10, BVFUNC_1:def 20; :: thesis: verum

end;

end;

end;

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