let Y be non empty set ; :: thesis: for a, b being Function of Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for PA being a_partition of Y holds All ((a 'eqv' b),PA,G) '<' (All (a,PA,G)) 'eqv' (All (b,PA,G))
let a, b be Function of Y,BOOLEAN; :: thesis: for G being Subset of (PARTITIONS Y)
for PA being a_partition of Y holds All ((a 'eqv' b),PA,G) '<' (All (a,PA,G)) 'eqv' (All (b,PA,G))
let G be Subset of (PARTITIONS Y); :: thesis: for PA being a_partition of Y holds All ((a 'eqv' b),PA,G) '<' (All (a,PA,G)) 'eqv' (All (b,PA,G))
let PA be a_partition of Y; :: thesis: All ((a 'eqv' b),PA,G) '<' (All (a,PA,G)) 'eqv' (All (b,PA,G))
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not (All ((a 'eqv' b),PA,G)) . z = TRUE or ((All (a,PA,G)) 'eqv' (All (b,PA,G))) . z = TRUE )
assume A1: (All ((a 'eqv' b),PA,G)) . z = TRUE ; :: thesis: ((All (a,PA,G)) 'eqv' (All (b,PA,G))) . z = TRUE
A2: ((All (a,PA,G)) 'eqv' (All (b,PA,G))) . z = 'not' (((All (a,PA,G)) . z) 'xor' ((All (b,PA,G)) . z)) by BVFUNC_1:def 9
.= ((((All (a,PA,G)) . z) 'or' ('not' ((All (b,PA,G)) . z))) '&' ('not' ((All (a,PA,G)) . z))) 'or' ((((All (a,PA,G)) . z) 'or' ('not' ((All (b,PA,G)) . z))) '&' ((All (b,PA,G)) . z)) by XBOOLEAN:8
.= ((('not' ((All (a,PA,G)) . z)) '&' ((All (a,PA,G)) . z)) 'or' (('not' ((All (a,PA,G)) . z)) '&' ('not' ((All (b,PA,G)) . z)))) 'or' (((All (b,PA,G)) . z) '&' (((All (a,PA,G)) . z) 'or' ('not' ((All (b,PA,G)) . z)))) by XBOOLEAN:8
.= ((('not' ((All (a,PA,G)) . z)) '&' ((All (a,PA,G)) . z)) 'or' (('not' ((All (a,PA,G)) . z)) '&' ('not' ((All (b,PA,G)) . z)))) 'or' ((((All (b,PA,G)) . z) '&' ((All (a,PA,G)) . z)) 'or' (((All (b,PA,G)) . z) '&' ('not' ((All (b,PA,G)) . z)))) by XBOOLEAN:8
.= (FALSE 'or' (('not' ((All (a,PA,G)) . z)) '&' ('not' ((All (b,PA,G)) . z)))) 'or' ((((All (b,PA,G)) . z) '&' ((All (a,PA,G)) . z)) 'or' (((All (b,PA,G)) . z) '&' ('not' ((All (b,PA,G)) . z)))) by XBOOLEAN:138
.= (('not' ((All (a,PA,G)) . z)) '&' ('not' ((All (b,PA,G)) . z))) 'or' ((((All (b,PA,G)) . z) '&' ((All (a,PA,G)) . z)) 'or' FALSE) by XBOOLEAN:138
.= (('not' ((All (a,PA,G)) . z)) '&' ('not' ((All (b,PA,G)) . z))) 'or' (((All (b,PA,G)) . z) '&' ((All (a,PA,G)) . z)) ;

per cases ( ( ( for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
a . x = TRUE ) & ( for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
b . x = TRUE ) ) or ( ( for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
a . x = TRUE ) & ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not b . x = TRUE ) ) or ( ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not a . x = TRUE ) & ( for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
b . x = TRUE ) ) or ( ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not a . x = TRUE ) & ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not b . x = TRUE ) ) ) ;

suppose A3: ( ( for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
a . x = TRUE ) & ( for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
b . x = TRUE ) ) ; :: thesis: ((All (a,PA,G)) 'eqv' (All (b,PA,G))) . z = TRUE

then (B_INF (b,(CompF (PA,G)))) . z = TRUE by BVFUNC_1:def 16;
hence ((All (a,PA,G)) 'eqv' (All (b,PA,G))) . z = TRUE by A2, A3, BVFUNC_1:def 16; :: thesis: verum

end;

suppose A4: ( ( for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
a . x = TRUE ) & ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not b . x = TRUE ) ) ; :: thesis: ((All (a,PA,G)) 'eqv' (All (b,PA,G))) . z = TRUE

then consider x1 being Element of Y such that
A5: x1 in EqClass (z,(CompF (PA,G))) and
A6: b . x1 <> TRUE ;
A7: a . x1 = TRUE by A4, A5;
(a 'eqv' b) . x1 = 'not' ((a . x1) 'xor' (b . x1)) by BVFUNC_1:def 9
.= FALSE by A6, A7, XBOOLEAN:def 3 ;
hence ((All (a,PA,G)) 'eqv' (All (b,PA,G))) . z = TRUE by A1, A5, BVFUNC_1:def 16; :: thesis: verum

end;

suppose A8: ( ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not a . x = TRUE ) & ( for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds
b . x = TRUE ) ) ; :: thesis: ((All (a,PA,G)) 'eqv' (All (b,PA,G))) . z = TRUE

then consider x1 being Element of Y such that
A9: x1 in EqClass (z,(CompF (PA,G))) and
A10: a . x1 <> TRUE ;
A11: b . x1 = TRUE by A8, A9;
(a 'eqv' b) . x1 = 'not' ((a . x1) 'xor' (b . x1)) by BVFUNC_1:def 9
.= FALSE by A10, A11, XBOOLEAN:def 3 ;
hence ((All (a,PA,G)) 'eqv' (All (b,PA,G))) . z = TRUE by A1, A9, BVFUNC_1:def 16; :: thesis: verum

end;

suppose A12: ( ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not a . x = TRUE ) & ex x being Element of Y st
( x in EqClass (z,(CompF (PA,G))) & not b . x = TRUE ) ) ; :: thesis: ((All (a,PA,G)) 'eqv' (All (b,PA,G))) . z = TRUE

then (B_INF (b,(CompF (PA,G)))) . z = FALSE by BVFUNC_1:def 16;
hence ((All (a,PA,G)) 'eqv' (All (b,PA,G))) . z = TRUE by A2, A12, BVFUNC_1:def 16; :: thesis: verum

end;

end;