let Y be non empty set ; :: thesis: for a, b being Element of Funcs 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 Element of Funcs 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 15 :: 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 12
.=
((((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 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) &
b . x1 <> TRUE )
;
A6:
a . x1 = TRUE
by A4, A5;
(a 'eqv' b) . x1 =
'not' ((a . x1) 'xor' (b . x1))
by BVFUNC_1:def 12
.=
FALSE
by A5, A6, XBOOLEAN:def 3
;
hence
((All a,PA,G) 'eqv' (All b,PA,G)) . z = TRUE
by A1, A5, BVFUNC_1:def 19;
:: thesis: verum end; suppose A7:
( 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 A8:
(
x1 in EqClass z,
(CompF PA,G) &
a . x1 <> TRUE )
;
A9:
b . x1 = TRUE
by A7, A8;
(a 'eqv' b) . x1 =
'not' ((a . x1) 'xor' (b . x1))
by BVFUNC_1:def 12
.=
FALSE
by A8, A9, XBOOLEAN:def 3
;
hence
((All a,PA,G) 'eqv' (All b,PA,G)) . z = TRUE
by A1, A8, BVFUNC_1:def 19;
:: thesis: verum end;