:: CQC_THE2 semantic presentation
begin
theorem
:: CQC_THE2:1
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
,
r
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) st
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
r
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
holds
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
r
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:2
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
,
r
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) st
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
r
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
holds
(
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
r
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:3
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
,
r
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) st
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
r
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
holds
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
r
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:4
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
,
r
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) st
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
r
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
holds
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
r
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:5
for
A
being ( ( ) ( )
QC-alphabet
)
for
s
being ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
for
y
,
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
s
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) ) )
)
: ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) iff (
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
s
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
<>
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ) ) ;
theorem
:: CQC_THE2:6
for
A
being ( ( ) ( )
QC-alphabet
)
for
s
being ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
for
y
,
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
s
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) ) )
)
: ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) iff (
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
s
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
<>
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ) ) ;
theorem
:: CQC_THE2:7
for
A
being ( ( ) ( )
QC-alphabet
)
for
s
,
h
being ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
for
y
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
(
s
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
=>
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) iff (
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
s
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) or
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) ) ) ;
theorem
:: CQC_THE2:8
for
A
being ( ( ) ( )
QC-alphabet
)
for
s
,
h
being ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
for
y
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
(
s
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
'&'
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) iff (
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
s
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) or
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) ) ) ;
theorem
:: CQC_THE2:9
for
A
being ( ( ) ( )
QC-alphabet
)
for
s
,
h
being ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
for
y
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
(
s
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
'or'
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) iff (
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
s
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) or
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) ) ) ;
theorem
:: CQC_THE2:10
for
A
being ( ( ) ( )
QC-alphabet
)
for
s
being ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
for
x
,
y
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
( not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
s
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) ) )
)
: ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) & not
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
s
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) ) )
)
: ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) ) ;
theorem
:: CQC_THE2:11
for
A
being ( ( ) ( )
QC-alphabet
)
for
s
being ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
for
x
,
y
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
( not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
s
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) ) )
)
: ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) & not
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
s
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) ) )
)
: ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) ) ;
theorem
:: CQC_THE2:12
for
A
being ( ( ) ( )
QC-alphabet
)
for
s
,
h
being ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
s
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
=>
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) )
.
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) )
=
(
s
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
.
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) )
=>
(
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
.
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) ;
theorem
:: CQC_THE2:13
for
A
being ( ( ) ( )
QC-alphabet
)
for
s
,
h
being ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
s
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
'or'
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) )
.
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) )
=
(
s
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
.
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) )
'or'
(
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
.
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) ;
theorem
:: CQC_THE2:14
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
,
y
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
<>
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
.
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) )
=
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
.
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) ) : ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) ;
theorem
:: CQC_THE2:15
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:16
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
holds
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:17
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:18
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
,
y
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:19
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
& not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:20
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:21
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
holds
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:22
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
h
being ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
for
x
,
y
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
.
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) &
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
.
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) & not
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:23
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
holds
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:24
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:25
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
h
being ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
for
x
,
y
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
.
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) &
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
.
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) & not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:26
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
y
,
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
All
(
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:27
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
h
being ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
for
x
,
y
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
.
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) &
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
.
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) & not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) & not
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
All
(
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:28
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
,
y
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:29
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
h
being ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
for
x
,
y
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
.
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) &
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
.
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) & not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) & not
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:30
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:31
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
holds
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:32
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:33
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
holds
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:34
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:35
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
holds
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:36
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
&
(
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
) ;
theorem
:: CQC_THE2:37
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=>
(
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:38
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
iff
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
) ;
theorem
:: CQC_THE2:39
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'or'
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'or'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:40
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'or'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'or'
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
&
(
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'or'
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'or'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
) ;
theorem
:: CQC_THE2:41
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'or'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=>
(
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'or'
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:42
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'or'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
iff
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'or'
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
) ;
theorem
:: CQC_THE2:43
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:44
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
holds
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:45
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
'not'
(
'not'
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
&
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
'not'
(
'not'
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
) ;
theorem
:: CQC_THE2:46
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
'not'
(
'not'
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:47
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
'not'
(
'not'
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
&
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
'not'
(
'not'
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
) ;
theorem
:: CQC_THE2:48
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
'not'
(
'not'
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:49
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
(
'not'
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
'not'
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
&
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
'not'
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
'not'
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
) ;
theorem
:: CQC_THE2:50
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
'not'
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
'not'
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:51
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
(
'not'
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
'not'
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
&
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
'not'
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
'not'
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
) ;
theorem
:: CQC_THE2:52
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
'not'
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
'not'
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:53
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
(
'not'
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
'not'
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
&
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
'not'
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
'not'
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
) ;
theorem
:: CQC_THE2:54
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
'not'
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=>
(
'not'
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:55
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
,
y
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
All
(
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
All
(
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
&
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) )
=>
(
All
(
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) is
valid
) ;
theorem
:: CQC_THE2:56
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
h
being ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
for
x
,
y
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
.
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) &
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
.
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) & not
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
All
(
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:57
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
,
y
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
Ex
(
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
&
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) )
=>
(
Ex
(
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) is
valid
) ;
theorem
:: CQC_THE2:58
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
h
being ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
for
x
,
y
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
.
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) &
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) )
.
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) & not
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
h
: ( ( ) ( )
QC-formula
of ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( non
empty
) ( non
empty
)
set
) ) is
valid
;
theorem
:: CQC_THE2:59
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
,
y
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
All
(
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
All
(
y
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:60
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=>
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:61
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
&
(
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
) ;
theorem
:: CQC_THE2:62
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=>
(
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:63
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
iff
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
) ;
theorem
:: CQC_THE2:64
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:65
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:66
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:67
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) &
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
holds
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:68
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'or'
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'or'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
&
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'or'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'or'
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
) ;
theorem
:: CQC_THE2:69
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'or'
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'or'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:70
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'or'
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
iff
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'or'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
) ;
theorem
:: CQC_THE2:71
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
&
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
) ;
theorem
:: CQC_THE2:72
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:73
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
iff
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
) ;
theorem
:: CQC_THE2:74
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
&
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
) ;
theorem
:: CQC_THE2:75
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:76
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
iff
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
) ;
theorem
:: CQC_THE2:77
for
A
being ( ( ) ( )
QC-alphabet
)
for
q
,
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:78
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:79
for
A
being ( ( ) ( )
QC-alphabet
)
for
q
,
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
iff
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
) ;
theorem
:: CQC_THE2:80
for
A
being ( ( ) ( )
QC-alphabet
)
for
q
,
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
(
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
&
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
) ;
theorem
:: CQC_THE2:81
for
A
being ( ( ) ( )
QC-alphabet
)
for
q
,
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:82
for
A
being ( ( ) ( )
QC-alphabet
)
for
q
,
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
iff
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
) ;
theorem
:: CQC_THE2:83
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:84
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) holds
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:85
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
;
theorem
:: CQC_THE2:86
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
iff
Ex
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
valid
) ;
theorem
:: CQC_THE2:87
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
{
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
}
: ( ( ) ( non
empty
)
Element
of
bool
(
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) )
|-
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: CQC_THE2:88
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
Cn
(
{
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
}
: ( ( ) ( non
empty
)
Element
of
bool
(
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) )
\/
{
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
}
: ( ( ) ( non
empty
)
Element
of
bool
(
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) )
)
: ( ( ) ( non
empty
)
Element
of
bool
(
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( )
Element
of
bool
(
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) )
=
Cn
{
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
}
: ( ( ) ( non
empty
)
Element
of
bool
(
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( )
Element
of
bool
(
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: CQC_THE2:89
for
A
being ( ( ) ( )
QC-alphabet
)
for
p
,
q
,
r
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
(
{
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
}
: ( ( ) ( non
empty
)
Element
of
bool
(
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) )
|-
r
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) iff
{
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
'&'
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
}
: ( ( ) ( non
empty
)
Element
of
bool
(
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) )
|-
r
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) ;
theorem
:: CQC_THE2:90
for
A
being ( ( ) ( )
QC-alphabet
)
for
X
being ( ( ) ( )
Subset
of ( ( ) ( )
set
) )
for
p
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st
X
: ( ( ) ( )
Subset
of ( ( ) ( )
set
) )
|-
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
X
: ( ( ) ( )
Subset
of ( ( ) ( )
set
) )
|-
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: CQC_THE2:91
for
A
being ( ( ) ( )
QC-alphabet
)
for
X
being ( ( ) ( )
Subset
of ( ( ) ( )
set
) )
for
p
,
q
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
for
x
being ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
A
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) st not
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) )
in
still_not-bound_in
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
bool
(
bound_QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
X
: ( ( ) ( )
Subset
of ( ( ) ( )
set
) )
|-
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
p
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
(
All
(
x
: ( ( ) ( )
bound_QC-variable
of ( ( ) ( )
Element
of
bool
(
QC-variables
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ) ,
q
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: CQC_THE2:92
for
A
being ( ( ) ( )
QC-alphabet
)
for
X
being ( ( ) ( )
Subset
of ( ( ) ( )
set
) )
for
F
,
G
being ( ( ) ( )
Element
of
CQC-WFF
A
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) st
F
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) is
closed
&
X
: ( ( ) ( )
Subset
of ( ( ) ( )
set
) )
\/
{
F
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
}
: ( ( ) ( non
empty
)
Element
of
bool
(
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
Element
of
bool
(
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) )
|-
G
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
X
: ( ( ) ( )
Subset
of ( ( ) ( )
set
) )
|-
F
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=>
G
: ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( )
Element
of
CQC-WFF
b
1
: ( ( ) ( )
QC-alphabet
) : ( ( ) ( non
empty
)
Element
of
bool
(
QC-WFF
b
1
: ( ( ) ( )
QC-alphabet
)
)
: ( ( non
empty
) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;