:: XREAL_1 semantic presentation
begin
theorem
:: XREAL_1:1
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) ex
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:2
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) ex
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:3
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) ex
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
(
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
) ) ;
theorem
:: XREAL_1:4
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) ex
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
(
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) ) ;
theorem
:: XREAL_1:5
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
ex
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
(
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) ) ;
begin
theorem
:: XREAL_1:6
for
a
,
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) holds
(
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) iff
a
: ( (
real
) (
complex
ext-real
real
)
number
)
+
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
)
+
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ) ;
theorem
:: XREAL_1:7
for
a
,
b
,
c
,
d
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
d
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
+
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
)
+
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:8
for
a
,
b
,
c
,
d
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
d
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
+
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
)
+
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
begin
theorem
:: XREAL_1:9
for
a
,
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) holds
(
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) iff
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ) ;
theorem
:: XREAL_1:10
for
a
,
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) holds
(
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) iff
c
: ( (
real
) (
complex
ext-real
real
)
number
)
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
)
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ) ;
theorem
:: XREAL_1:11
for
a
,
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:12
for
a
,
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:13
for
a
,
b
,
c
,
d
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
d
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:14
for
a
,
b
,
c
,
d
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
d
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:15
for
a
,
b
,
c
,
d
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<
d
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:16
for
a
,
b
,
c
,
d
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
)
-
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:17
for
a
,
b
,
c
,
d
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
)
-
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
d
: ( (
real
) (
complex
ext-real
real
)
number
)
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
)
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:18
for
a
,
b
,
c
,
d
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
)
-
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
d
: ( (
real
) (
complex
ext-real
real
)
number
)
-
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
begin
theorem
:: XREAL_1:19
for
a
,
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) holds
(
a
: ( (
real
) (
complex
ext-real
real
)
number
)
+
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) iff
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
)
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ) ;
theorem
:: XREAL_1:20
for
a
,
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) holds
(
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
)
+
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) iff
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) ) ;
theorem
:: XREAL_1:21
for
a
,
b
,
c
,
d
being ( (
real
) (
complex
ext-real
real
)
number
) holds
(
a
: ( (
real
) (
complex
ext-real
real
)
number
)
+
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
)
+
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) iff
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
d
: ( (
real
) (
complex
ext-real
real
)
number
)
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ) ;
theorem
:: XREAL_1:22
for
a
,
b
,
c
,
d
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
+
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
)
-
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
+
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
)
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:23
for
a
,
b
,
c
,
d
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
)
+
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
)
+
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
begin
theorem
:: XREAL_1:24
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) holds
(
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) iff
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
)
<=
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
) ) ;
theorem
:: XREAL_1:25
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:26
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) ;
begin
theorem
:: XREAL_1:27
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
+
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:28
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
+
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
begin
theorem
:: XREAL_1:29
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
)
+
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:30
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
+
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:31
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
+
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:32
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
+
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) ;
begin
theorem
:: XREAL_1:33
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
+
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:34
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
+
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:35
for
a
,
c
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
+
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:36
for
a
,
c
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
+
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:37
for
a
,
c
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
+
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:38
for
a
,
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
+
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:39
for
a
,
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
+
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:40
for
a
,
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
+
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
begin
theorem
:: XREAL_1:41
for
c
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st ( for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
>
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
)
+
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:42
for
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st ( for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
+
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) ) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) ;
begin
theorem
:: XREAL_1:43
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:44
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:45
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:46
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:47
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:48
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:49
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:50
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:51
for
a
,
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:52
for
a
,
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
)
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:53
for
a
,
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
)
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:54
for
a
,
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
)
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:55
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) holds
( not
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<>
b
: ( (
real
) (
complex
ext-real
real
)
number
) or
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) or
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ) ;
begin
theorem
:: XREAL_1:56
for
c
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st ( for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
>=
c
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:57
for
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st ( for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
>
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) ) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) ;
begin
theorem
:: XREAL_1:58
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) holds
(
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) iff
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
) ) ;
theorem
:: XREAL_1:59
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
+
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:60
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
+
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:61
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
+
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:62
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
+
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
begin
theorem
:: XREAL_1:63
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) holds
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:64
for
a
,
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
)
*
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:65
for
a
,
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
*
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:66
for
a
,
b
,
c
,
d
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
d
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
)
*
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:67
for
a
,
b
,
c
,
d
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
d
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
)
*
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:68
for
c
,
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
c
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
)
*
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:69
for
c
,
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
*
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:70
for
a
,
b
,
c
,
d
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
d
: ( (
real
) (
complex
ext-real
real
)
number
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
)
*
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
begin
theorem
:: XREAL_1:71
for
a
,
b
,
c
,
d
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
d
: ( (
real
) (
complex
ext-real
real
)
number
) &
(
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
c
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
+
(
b
: ( (
real
) (
complex
ext-real
real
)
number
)
*
d
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
) : ( ( ) (
complex
ext-real
real
)
set
)
=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) & not
a
: ( (
real
) (
complex
ext-real
real
)
number
)
=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
begin
theorem
:: XREAL_1:72
for
c
,
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
/
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:73
for
c
,
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
/
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:74
for
c
,
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
c
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
)
/
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:75
for
c
,
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
/
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:76
for
c
,
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
c
: ( (
real
) (
complex
ext-real
real
)
number
) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
begin
theorem
:: XREAL_1:77
for
b
,
a
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:78
for
b
,
a
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:79
for
b
,
c
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:80
for
b
,
c
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:81
for
b
,
a
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:82
for
b
,
a
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:83
for
b
,
c
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:84
for
b
,
c
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
begin
theorem
:: XREAL_1:85
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:86
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:87
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:88
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:89
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:90
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:91
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:92
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) ;
begin
theorem
:: XREAL_1:93
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
(
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-
a
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
*
(
b
: ( (
real
) (
complex
ext-real
real
)
number
)
+
a
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
(
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) ) ;
theorem
:: XREAL_1:94
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
(
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-
a
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
*
(
b
: ( (
real
) (
complex
ext-real
real
)
number
)
+
a
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
) : ( ( ) (
complex
ext-real
real
)
set
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
(
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) ) ;
theorem
:: XREAL_1:95
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) holds
( not
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
(
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-
a
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
*
(
b
: ( (
real
) (
complex
ext-real
real
)
number
)
+
a
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
) : ( ( ) (
complex
ext-real
real
)
set
) or
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
) or
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) ) ;
theorem
:: XREAL_1:96
for
a
,
c
,
b
,
d
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<
d
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
)
*
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:97
for
a
,
c
,
b
,
d
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
c
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
d
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
)
*
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:98
for
a
,
c
,
b
,
d
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
c
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<
d
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
)
*
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:99
for
c
,
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
c
: ( (
real
) (
complex
ext-real
real
)
number
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:100
for
c
,
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:101
for
c
,
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:102
for
b
,
d
,
a
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
d
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:103
for
b
,
d
,
a
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
d
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:104
for
b
,
d
,
a
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
d
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:105
for
b
,
d
,
a
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
d
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:106
for
b
,
d
,
a
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
d
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:107
for
b
,
d
,
a
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
d
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:108
for
b
,
d
,
a
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
d
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:109
for
b
,
d
,
a
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
d
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:110
for
b
,
d
,
a
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
d
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:111
for
b
,
d
,
a
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
d
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:112
for
b
,
d
,
c
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
d
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:113
for
b
,
d
,
c
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
d
: ( (
real
) (
complex
ext-real
real
)
number
) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:114
for
b
,
d
,
a
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
d
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:115
for
b
,
d
,
a
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
d
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:116
for
b
,
d
,
c
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
d
: ( (
real
) (
complex
ext-real
real
)
number
) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:117
for
b
,
d
,
c
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
d
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
d
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:118
for
a
,
c
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:119
for
c
,
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:120
for
c
,
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:121
for
c
,
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:122
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) holds
(
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) iff
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
) ) ;
theorem
:: XREAL_1:123
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) holds
(
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) iff
a
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) ;
theorem
:: XREAL_1:124
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:125
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) ;
begin
theorem
:: XREAL_1:126
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:127
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:128
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:129
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:130
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:131
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:132
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:133
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) holds
( not
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) or (
a
: ( (
real
) (
complex
ext-real
real
)
number
)
>
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) or (
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
>
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) ) ;
theorem
:: XREAL_1:134
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) holds
( not
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
>
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) or (
a
: ( (
real
) (
complex
ext-real
real
)
number
)
>
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
>
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) or (
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) ) ;
theorem
:: XREAL_1:135
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:136
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:137
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:138
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:139
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:140
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:141
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:142
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
/
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:143
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) holds
( not
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) or (
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
>
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) or (
b
: ( (
real
) (
complex
ext-real
real
)
number
)
>
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) ) ;
theorem
:: XREAL_1:144
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) holds
( not
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
>
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) or (
b
: ( (
real
) (
complex
ext-real
real
)
number
)
>
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
>
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) or (
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) ) ;
begin
theorem
:: XREAL_1:145
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
)
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:146
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:147
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
complex
ext-real
real
)
set
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:148
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
+
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:149
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
>
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
begin
theorem
:: XREAL_1:150
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:151
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) & 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:152
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:153
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:154
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) & 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:155
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) & 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:156
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:157
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:158
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) & 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:159
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) & 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:160
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:161
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) & 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:162
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:163
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
) holds
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:164
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
) holds
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:165
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:166
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
begin
theorem
:: XREAL_1:167
for
c
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st ( for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
)
*
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:168
for
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st ( for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
*
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) ) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:169
for
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st ( for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
*
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:170
for
d
,
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
d
: ( (
real
) (
complex
ext-real
real
)
number
) &
d
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
(
d
: ( (
real
) (
complex
ext-real
real
)
number
)
*
a
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
+
(
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
-
d
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
) : ( ( ) (
complex
ext-real
real
)
set
)
=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) & not (
d
: ( (
real
) (
complex
ext-real
real
)
number
)
=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) & not (
d
: ( (
real
) (
complex
ext-real
real
)
number
)
=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) holds
(
a
: ( (
real
) (
complex
ext-real
real
)
number
)
=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ) ;
theorem
:: XREAL_1:171
for
d
,
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
d
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
(
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
-
d
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
*
a
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
+
(
d
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:172
for
d
,
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
d
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
(
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
-
d
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
*
a
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
+
(
d
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:173
for
d
,
a
,
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
d
: ( (
real
) (
complex
ext-real
real
)
number
) &
d
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
(
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
-
d
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
+
(
d
: ( (
real
) (
complex
ext-real
real
)
number
)
*
c
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:174
for
d
,
b
,
a
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
d
: ( (
real
) (
complex
ext-real
real
)
number
) &
d
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
(
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
-
d
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
+
(
d
: ( (
real
) (
complex
ext-real
real
)
number
)
*
c
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:175
for
d
,
a
,
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
d
: ( (
real
) (
complex
ext-real
real
)
number
) &
d
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
(
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
-
d
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
+
(
d
: ( (
real
) (
complex
ext-real
real
)
number
)
*
c
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:176
for
d
,
b
,
a
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
d
: ( (
real
) (
complex
ext-real
real
)
number
) &
d
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
(
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
-
d
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
+
(
d
: ( (
real
) (
complex
ext-real
real
)
number
)
*
c
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
) : ( ( ) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:177
for
d
,
a
,
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
d
: ( (
real
) (
complex
ext-real
real
)
number
) &
d
: ( (
real
) (
complex
ext-real
real
)
number
)
<
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
(
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
-
d
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
+
(
d
: ( (
real
) (
complex
ext-real
real
)
number
)
*
c
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:178
for
d
,
b
,
a
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
d
: ( (
real
) (
complex
ext-real
real
)
number
) &
d
: ( (
real
) (
complex
ext-real
real
)
number
)
<
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
(
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
-
d
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
+
(
d
: ( (
real
) (
complex
ext-real
real
)
number
)
*
c
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
) : ( ( ) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:179
for
d
,
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
d
: ( (
real
) (
complex
ext-real
real
)
number
) &
d
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
(
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
-
d
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
*
a
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
+
(
d
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
) : ( ( ) (
complex
ext-real
real
)
set
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
(
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
-
d
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
*
a
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
+
(
d
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
) : ( ( ) (
complex
ext-real
real
)
set
) holds
d
: ( (
real
) (
complex
ext-real
real
)
number
)
=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:180
for
d
,
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
d
: ( (
real
) (
complex
ext-real
real
)
number
) &
d
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
(
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
-
d
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
*
a
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
+
(
d
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
(
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
-
d
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
*
a
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
+
(
d
: ( (
real
) (
complex
ext-real
real
)
number
)
*
b
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
) : ( ( ) (
complex
ext-real
real
)
set
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
d
: ( (
real
) (
complex
ext-real
real
)
number
)
=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
begin
theorem
:: XREAL_1:181
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
)
/
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:182
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
)
/
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:183
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:184
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:185
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
/
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:186
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
/
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:187
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
)
/
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:188
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
)
/
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:189
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:190
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:191
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
/
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:192
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
/
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
begin
theorem
:: XREAL_1:193
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:194
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:195
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
) holds
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:196
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
) holds
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:197
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:198
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:199
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
) holds
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:200
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
) holds
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:201
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
) ;
theorem
:: XREAL_1:202
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
) ;
theorem
:: XREAL_1:203
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
) ;
theorem
:: XREAL_1:204
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
) ;
theorem
:: XREAL_1:205
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
) ;
theorem
:: XREAL_1:206
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
) ;
theorem
:: XREAL_1:207
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
) ;
theorem
:: XREAL_1:208
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) &
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( (
complex
) (
complex
ext-real
real
)
set
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
) ;
begin
theorem
:: XREAL_1:209
for
c
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st ( for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
)
/
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ) holds
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:210
for
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st ( for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
/
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) ) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:211
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
)
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:212
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
)
<
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:213
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
) holds
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:214
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
) holds
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( (
complex
) ( non
empty
complex
ext-real
non
positive
negative
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
"
: ( (
complex
) (
complex
ext-real
real
)
set
) ;
begin
theorem
:: XREAL_1:215
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:216
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:217
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
complex
ext-real
positive
non
negative
real
)
set
) holds
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
*
a
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
complex
ext-real
real
)
set
)
<=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:218
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
complex
ext-real
positive
non
negative
real
)
set
) holds
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
-
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
*
a
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:219
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
>=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
complex
ext-real
positive
non
negative
real
)
set
) holds
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
*
a
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
complex
ext-real
real
)
set
)
>=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:220
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
>=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
complex
ext-real
positive
non
negative
real
)
set
) holds
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
>=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
-
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
*
a
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
) : ( ( ) (
complex
ext-real
real
)
set
) ;
begin
theorem
:: XREAL_1:221
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:222
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
complex
ext-real
real
)
set
) ;
begin
theorem
:: XREAL_1:223
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
complex
ext-real
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:224
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
)
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:225
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<>
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) holds
ex
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
=
b
: ( (
real
) (
complex
ext-real
real
)
number
)
/
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
begin
theorem
:: XREAL_1:226
for
r
,
s
being ( (
real
) (
complex
ext-real
real
)
number
) st
r
: ( (
real
) (
complex
ext-real
real
)
number
)
<
s
: ( (
real
) (
complex
ext-real
real
)
number
) holds
(
r
: ( (
real
) (
complex
ext-real
real
)
number
)
<
(
r
: ( (
real
) (
complex
ext-real
real
)
number
)
+
s
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
complex
ext-real
real
)
set
) &
(
r
: ( (
real
) (
complex
ext-real
real
)
number
)
+
s
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
real
)
set
)
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
complex
ext-real
real
)
set
)
<
s
: ( (
real
) (
complex
ext-real
real
)
number
) ) ;
registration
cluster
->
ext-real
for ( ( ) ( )
Element
of
REAL
: ( ( ) ( non
empty
)
set
) ) ;
end;
theorem
:: XREAL_1:227
for
r
,
t
being ( (
ext-real
) (
ext-real
)
number
) st
r
: ( (
ext-real
) (
ext-real
)
number
)
<
t
: ( (
ext-real
) (
ext-real
)
number
) holds
ex
s
being ( (
ext-real
) (
ext-real
)
number
) st
(
r
: ( (
ext-real
) (
ext-real
)
number
)
<
s
: ( (
ext-real
) (
ext-real
)
number
) &
s
: ( (
ext-real
) (
ext-real
)
number
)
<
t
: ( (
ext-real
) (
ext-real
)
number
) ) ;
theorem
:: XREAL_1:228
for
r
,
s
,
t
being ( (
ext-real
) (
ext-real
)
number
) st
r
: ( (
ext-real
) (
ext-real
)
number
)
<
s
: ( (
ext-real
) (
ext-real
)
number
) & ( for
q
being ( (
ext-real
) (
ext-real
)
number
) st
r
: ( (
ext-real
) (
ext-real
)
number
)
<
q
: ( (
ext-real
) (
ext-real
)
number
) &
q
: ( (
ext-real
) (
ext-real
)
number
)
<
s
: ( (
ext-real
) (
ext-real
)
number
) holds
t
: ( (
ext-real
) (
ext-real
)
number
)
<=
q
: ( (
ext-real
) (
ext-real
)
number
) ) holds
t
: ( (
ext-real
) (
ext-real
)
number
)
<=
r
: ( (
ext-real
) (
ext-real
)
number
) ;
theorem
:: XREAL_1:229
for
r
,
s
,
t
being ( (
ext-real
) (
ext-real
)
number
) st
r
: ( (
ext-real
) (
ext-real
)
number
)
<
s
: ( (
ext-real
) (
ext-real
)
number
) & ( for
q
being ( (
ext-real
) (
ext-real
)
number
) st
r
: ( (
ext-real
) (
ext-real
)
number
)
<
q
: ( (
ext-real
) (
ext-real
)
number
) &
q
: ( (
ext-real
) (
ext-real
)
number
)
<
s
: ( (
ext-real
) (
ext-real
)
number
) holds
q
: ( (
ext-real
) (
ext-real
)
number
)
<=
t
: ( (
ext-real
) (
ext-real
)
number
) ) holds
s
: ( (
ext-real
) (
ext-real
)
number
)
<=
t
: ( (
ext-real
) (
ext-real
)
number
) ;
theorem
:: XREAL_1:230
for
a
,
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:231
for
a
,
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
c
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
<
c
: ( (
real
) (
complex
ext-real
real
)
number
) ;
begin
theorem
:: XREAL_1:232
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-'
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
non
negative
real
)
set
)
=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
complex
ext-real
non
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) ;
theorem
:: XREAL_1:233
for
b
,
a
being ( (
real
) (
complex
ext-real
real
)
number
) st
b
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-'
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
non
negative
real
)
set
)
=
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
) ;
theorem
:: XREAL_1:234
for
c
,
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-'
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
non
negative
real
)
set
)
=
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-'
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
non
negative
real
)
set
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
=
b
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:235
for
a
,
b
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
>=
b
: ( (
real
) (
complex
ext-real
real
)
number
) holds
(
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-'
b
: ( (
real
) (
complex
ext-real
real
)
number
)
)
: ( ( ) (
complex
ext-real
non
negative
real
)
set
)
+
b
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
real
)
set
)
=
a
: ( (
real
) (
complex
ext-real
real
)
number
) ;
theorem
:: XREAL_1:236
for
a
,
b
,
c
being ( (
real
) (
complex
ext-real
real
)
number
) st
a
: ( (
real
) (
complex
ext-real
real
)
number
)
<=
b
: ( (
real
) (
complex
ext-real
real
)
number
) &
c
: ( (
real
) (
complex
ext-real
real
)
number
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-'
a
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
non
negative
real
)
set
)
<
b
: ( (
real
) (
complex
ext-real
real
)
number
)
-'
c
: ( (
real
) (
complex
ext-real
real
)
number
) : ( ( ) (
complex
ext-real
non
negative
real
)
set
) ;
theorem
:: XREAL_1:237
for
a
being ( (
real
) (
complex
ext-real
real
)
number
) st 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) )
<=
a
: ( (
real
) (
complex
ext-real
real
)
number
) holds
a
: ( (
real
) (
complex
ext-real
real
)
number
)
-'
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
complex
ext-real
positive
non
negative
real
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
)
set
) : ( ( ) ( )
set
) ) ) : ( ( ) (
complex
ext-real
non
negative
real
)
set
)
<
a
: ( (
real
) (
complex
ext-real
real
)
number
) ;