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 )
assume A1: ((Ex a,PA,G) 'xor' (Ex b,PA,G)) . z = TRUE ; :: thesis: (Ex (a 'xor' b),PA,G) . z = TRUE
A2: ((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))) ;
A3: ( ('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;
A4: ( 'not' FALSE = TRUE & 'not' TRUE = FALSE ) by MARGREL1:41;

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 A1, A2, A3, BINARITH:7;

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

then A5: ( 'not' ((Ex a,PA,G) . z) = TRUE & (Ex b,PA,G) . z = TRUE ) by MARGREL1:45;
then A6: (Ex a,PA,G) . z = FALSE by MARGREL1:41;
consider x1 being Element of Y such that
A7: ( x1 in EqClass z,(CompF PA,G) & b . x1 = TRUE ) by A5, BVFUNC_1:def 20;
a . x1 <> TRUE by A6, A7, 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 A4, A7, A8
.= TRUE by BINARITH:19 ;
hence (Ex (a 'xor' b),PA,G) . z = TRUE by A7, BVFUNC_1:def 20; :: thesis: verum

end;

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

then A9: ( (Ex a,PA,G) . z = TRUE & 'not' ((Ex b,PA,G) . z) = TRUE ) by MARGREL1:45;
then A10: (Ex b,PA,G) . z = FALSE by MARGREL1:41;
consider x1 being Element of Y such that
A11: ( x1 in EqClass z,(CompF PA,G) & a . x1 = TRUE ) by A9, BVFUNC_1:def 20;
b . x1 <> TRUE by A10, A11, 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 A4, A11, A12
.= TRUE by BINARITH:19 ;
hence (Ex (a 'xor' b),PA,G) . z = TRUE by A11, BVFUNC_1:def 20; :: thesis: verum

end;

end;

end;

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