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 All (a 'or' b),PA,G '<' (Ex a,PA,G) 'or' (All 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 All (a 'or' b),PA,G '<' (Ex a,PA,G) 'or' (All b,PA,G)
let a, b be Element of Funcs Y,BOOLEAN ; :: thesis: for PA being a_partition of Y holds All (a 'or' b),PA,G '<' (Ex a,PA,G) 'or' (All b,PA,G)
let PA be a_partition of Y; :: thesis: All (a 'or' b),PA,G '<' (Ex a,PA,G) 'or' (All b,PA,G)
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not (All (a 'or' b),PA,G) . z = TRUE or ((Ex a,PA,G) 'or' (All b,PA,G)) . z = TRUE )
assume A1:
(All (a 'or' b),PA,G) . z = TRUE
; :: thesis: ((Ex a,PA,G) 'or' (All b,PA,G)) . z = TRUE
per cases
( ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & a . x = TRUE ) or ( ( for x being Element of Y st x in EqClass z,(CompF PA,G) holds
b . x = TRUE ) & ( for x being Element of Y holds
( not x in EqClass z,(CompF PA,G) or not a . x = TRUE ) ) ) or ( ex x being Element of Y st
( x in EqClass z,(CompF PA,G) & not b . x = TRUE ) & ( for x being Element of Y holds
( not x in EqClass z,(CompF PA,G) or not a . x = TRUE ) ) ) )
;
suppose
ex
x being
Element of
Y st
(
x in EqClass z,
(CompF PA,G) &
a . x = TRUE )
;
:: thesis: ((Ex a,PA,G) 'or' (All b,PA,G)) . z = TRUE then
(B_SUP a,(CompF PA,G)) . z = TRUE
by BVFUNC_1:def 20;
then
(Ex a,PA,G) . z = TRUE
by BVFUNC_2:def 10;
hence ((Ex a,PA,G) 'or' (All b,PA,G)) . z =
TRUE 'or' ((All b,PA,G) . z)
by BVFUNC_1:def 7
.=
TRUE
by BINARITH:19
;
:: thesis: verum end; suppose
( ( for
x being
Element of
Y st
x in EqClass z,
(CompF PA,G) holds
b . x = TRUE ) & ( for
x being
Element of
Y holds
( not
x in EqClass z,
(CompF PA,G) or not
a . x = TRUE ) ) )
;
:: thesis: ((Ex a,PA,G) 'or' (All b,PA,G)) . z = TRUE then
(B_INF b,(CompF PA,G)) . z = TRUE
by BVFUNC_1:def 19;
then
(All b,PA,G) . z = TRUE
by BVFUNC_2:def 9;
hence ((Ex a,PA,G) 'or' (All b,PA,G)) . z =
((Ex a,PA,G) . z) 'or' TRUE
by BVFUNC_1:def 7
.=
TRUE
by BINARITH:19
;
:: thesis: verum end; suppose A2:
( ex
x being
Element of
Y st
(
x in EqClass z,
(CompF PA,G) & not
b . x = TRUE ) & ( for
x being
Element of
Y holds
( not
x in EqClass z,
(CompF PA,G) or not
a . x = TRUE ) ) )
;
:: thesis: ((Ex a,PA,G) 'or' (All b,PA,G)) . z = TRUE then consider x1 being
Element of
Y such that A3:
(
x1 in EqClass z,
(CompF PA,G) &
b . x1 <> TRUE )
;
A4:
a . x1 <> TRUE
by A2, A3;
A5:
b . x1 = FALSE
by A3, XBOOLEAN:def 3;
(a 'or' b) . x1 =
(a . x1) 'or' (b . x1)
by BVFUNC_1:def 7
.=
FALSE 'or' FALSE
by A4, A5, XBOOLEAN:def 3
.=
FALSE
;
hence
((Ex a,PA,G) 'or' (All b,PA,G)) . z = TRUE
by A1, A3, Lm1;
:: thesis: verum end;