let Y be non empty set ; for G being Subset of (PARTITIONS Y)
for a, b being Function of 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); for a, b being Function of 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 Function of 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 PA be a_partition of Y; All ((a 'or' b),PA,G) '<' (Ex (a,PA,G)) 'or' (All (b,PA,G))
let z be Element of Y; BVFUNC_1:def 12 ( 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
; ((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 )
;
((Ex (a,PA,G)) 'or' (All (b,PA,G))) . z = TRUE then
(B_SUP (a,(CompF (PA,G)))) . z = TRUE
by BVFUNC_1:def 17;
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 4
.=
TRUE
by BINARITH:10
;
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 ) ) )
;
((Ex (a,PA,G)) 'or' (All (b,PA,G))) . z = TRUE then
(B_INF (b,(CompF (PA,G)))) . z = TRUE
by BVFUNC_1:def 16;
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 4
.=
TRUE
by BINARITH:10
;
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 ) ) )
;
((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)))
and A4:
b . x1 <> TRUE
;
A5:
a . x1 <> TRUE
by A2, A3;
A6:
b . x1 = FALSE
by A4, XBOOLEAN:def 3;
(a 'or' b) . x1 =
(a . x1) 'or' (b . x1)
by BVFUNC_1:def 4
.=
FALSE 'or' FALSE
by A5, A6, XBOOLEAN:def 3
.=
FALSE
;
hence
((Ex (a,PA,G)) 'or' (All (b,PA,G))) . z = TRUE
by A1, A3, Lm1;
verum end; end;