Proof.
Suppose
x
1
,
x
2
,
…
,
x
2
k
are arbitrary points in
X
. From (2.1.1), define
(4)
y
2
k
+
2
n
-
1
=
S
x
2
n
-
1
,
x
2
n
,
…
,
x
2
k
+
2
n
-
2
=
g
x
2
k
+
2
n
-
1
y
2
k
+
2
n
=
T
x
2
n
,
x
2
n
+
1
,
…
,
x
2
k
+
2
n
-
1
=
f
x
2
k
+
2
n
for
n
=
1,2
,
…
.
Let
(5)
α
2
n
=
d
f
x
2
n
,
g
x
2
n
+
1
,
α
2
n
-
1
=
d
g
x
2
n
-
1
,
f
x
2
n
,
n
=
1,2
,
…
.
Let
θ
=
λ
1
/
2
k
and
μ
=
m
a
x
{
α
1
/
θ
,
α
2
/
θ
2
,
…
,
α
2
k
/
θ
2
k
}
.
Then
0
<
θ
<
1
and by selection of
μ
we have
(6)
α
n
≤
μ
θ
n
,
for
n
=
1,2
,
…
,
2
k
.
(7)
α
2
k
+
1
=
d
g
x
2
k
+
1
,
f
x
2
k
+
2
=
d
S
x
1
,
x
2
,
x
3
,
x
4
,
…
,
x
2
k
-
1
,
x
2
k
,
T
x
2
,
x
3
,
x
4
,
…
,
x
2
k
,
x
2
k
+
1
≤
λ
max
d
g
x
1
,
f
x
2
,
d
f
x
2
,
g
x
3
,
d
g
x
3
,
f
x
4
,
d
f
x
4
,
g
x
5
,
…
,
d
g
x
2
k
-
1
,
f
x
2
k
,
d
f
x
2
k
,
g
x
2
k
+
1
from
2.1
.
2
=
λ
max
α
1
,
α
2
,
α
3
,
α
4
,
…
,
α
2
k
-
1
,
α
2
k
≤
λ
max
μ
θ
,
μ
θ
2
,
μ
θ
3
,
μ
θ
4
,
…
,
μ
θ
2
k
-
1
,
μ
θ
2
k
from
6
=
λ
μ
θ
=
θ
2
k
μ
θ
=
μ
θ
2
k
+
1
,
(8)
α
2
k
+
2
=
d
f
x
2
k
+
2
,
g
x
2
k
+
3
=
d
S
x
3
,
x
4
,
x
5
,
x
6
,
…
,
x
2
k
+
1
,
x
2
k
+
2
,
T
x
2
,
x
3
,
x
4
,
…
,
x
2
k
,
x
2
k
+
1
≤
λ
max
d
g
x
3
,
f
x
2
,
d
f
x
4
,
g
x
3
,
d
g
x
5
,
f
x
4
,
d
f
x
6
,
g
x
5
,
…
,
d
g
x
2
k
+
1
,
f
x
2
k
,
d
f
x
2
k
+
2
,
g
x
2
k
+
1
from
2.1
.
2
=
λ
max
α
2
,
α
3
,
α
4
,
α
5
,
…
,
α
2
k
,
α
2
k
+
1
≤
λ
max
μ
θ
2
,
μ
θ
3
,
μ
θ
4
,
μ
θ
5
,
…
,
μ
θ
2
k
,
μ
θ
2
k
+
1
from
6
,
7
=
λ
μ
θ
2
=
θ
2
k
μ
θ
2
=
μ
θ
2
k
+
2
.
Continuing in this way, we get
(9)
α
n
≤
μ
θ
n
for
n
=
1,2
,
…
.
Now consider
(10)
d
y
2
k
+
2
n
-
1
,
y
2
k
+
2
n
=
d
S
x
2
n
-
1
,
x
2
n
,
x
2
n
+
1
,
x
2
n
+
2
,
…
,
x
2
k
+
2
n
-
3
,
x
2
k
+
2
n
-
2
,
T
x
2
n
,
x
2
n
+
1
,
x
2
n
+
2
,
x
2
n
+
3
,
…
,
x
2
k
+
2
n
-
2
,
x
2
k
+
2
n
-
1
≤
λ
max
d
g
x
2
n
-
1
,
f
x
2
n
,
d
f
x
2
n
,
g
x
2
n
+
1
,
d
g
x
2
n
+
1
,
f
x
2
n
+
2
,
d
f
x
2
n
+
2
,
g
x
2
n
+
3
,
…
,
d
g
x
2
k
+
2
n
-
3
,
f
x
2
k
+
2
n
-
2
,
d
f
x
2
k
+
2
n
-
2
,
g
x
2
k
+
2
n
-
1
=
λ
max
α
2
n
-
1
,
α
2
n
,
α
2
n
+
1
,
α
2
n
+
2
,
…
,
α
2
k
+
2
n
-
3
,
α
2
k
+
2
n
-
2
≤
λ
max
μ
θ
2
n
-
1
,
μ
θ
2
n
,
μ
θ
2
n
+
1
,
μ
θ
2
n
+
2
,
…
,
μ
θ
2
k
+
2
n
-
3
,
μ
θ
2
k
+
2
n
-
2
from
9
=
λ
μ
θ
2
n
-
1
=
θ
2
k
μ
θ
2
n
-
1
=
μ
θ
2
k
+
2
n
-
1
.
Also
(11)
d
y
2
k
+
2
n
,
y
2
k
+
2
n
+
1
=
d
S
x
2
n
+
1
,
x
2
n
+
2
,
x
2
n
+
3
,
x
2
n
+
4
,
…
,
x
2
k
+
2
n
-
1
,
x
2
k
+
2
n
,
T
x
2
n
,
x
2
n
+
1
,
x
2
n
+
2
,
x
2
n
+
3
,
…
,
x
2
k
+
2
n
-
2
,
x
2
k
+
2
n
-
1
≤
λ
max
d
g
x
2
n
+
1
,
f
x
2
n
,
d
f
x
2
n
+
2
,
g
x
2
n
+
1
,
d
g
x
2
n
+
3
,
f
x
2
n
+
2
,
d
f
x
2
n
+
4
,
g
x
2
n
+
3
,
…
,
d
g
x
2
k
+
2
n
-
1
,
f
x
2
k
+
2
n
-
2
,
d
f
x
2
k
+
2
n
,
g
x
2
k
+
2
n
-
1
=
λ
max
α
2
n
,
α
2
n
+
1
,
α
2
n
+
2
,
α
2
n
+
3
,
…
,
α
2
k
+
2
n
-
2
,
α
2
k
+
2
n
-
1
≤
λ
max
μ
θ
2
n
,
μ
θ
2
n
+
1
,
μ
θ
2
n
+
2
,
μ
θ
2
n
+
3
,
…
,
μ
θ
2
k
+
2
n
-
2
,
μ
θ
2
k
+
2
n
-
1
from
9
=
λ
μ
θ
2
n
=
θ
2
k
μ
θ
2
n
=
μ
θ
2
k
+
2
n
.
Thus from (10) and (11) we have
(12)
d
y
2
k
+
n
,
y
2
k
+
n
+
1
≤
μ
θ
2
k
+
n
for
n
=
1,2
,
3
,
…
.
Now from
m
>
n
consider
(13)
d
y
2
k
+
n
,
y
2
k
+
m
≤
d
y
2
k
+
n
,
y
2
k
+
n
+
1
+
d
y
2
k
+
n
+
1
,
y
2
k
+
n
+
2
+
d
y
2
k
+
n
+
2
,
y
2
k
+
n
+
3
+
⋯
+
d
y
2
k
+
m
-
1
,
y
2
k
+
m
≤
μ
θ
2
k
+
n
+
μ
θ
2
k
+
n
+
1
+
μ
θ
2
k
+
n
+
2
+
⋯
+
μ
θ
2
k
+
m
-
1
from
12
≤
μ
θ
2
k
θ
n
1
-
θ
⟶
0
as
n
⟶
∞
,
m
⟶
∞
.
Hence
{
y
2
k
+
n
}
is a Cauchy sequence in
X
. Since
X
is complete there exists
z
∈
X
such that
y
2
k
+
n
→
z
as
n
→
∞
. From (2.1.4), there exists
u
∈
X
such that
(14)
z
=
f
u
=
g
u
.
Now consider
(15)
d
S
u
,
u
,
…
,
u
,
u
,
y
2
k
+
2
n
=
d
S
u
,
u
,
…
,
u
,
u
,
T
x
2
n
,
x
2
n
+
1
,
…
,
x
2
k
+
2
n
-
2
,
x
2
k
+
2
n
-
3
≤
λ
max
d
g
p
,
f
x
2
n
,
d
f
p
,
g
x
2
n
+
1
,
d
g
p
,
f
x
2
n
+
2
,
d
f
p
,
g
x
2
n
+
3
,
…
,
d
g
p
,
f
x
2
k
+
2
n
-
2
,
d
f
p
,
g
x
2
k
+
2
n
-
1
.
Letting
n
→
∞
and using (14)
d
(
S
(
u
,
u
,
…
,
u
,
u
)
,
f
u
)
≤
0
so that
(16)
f
u
=
S
u
,
u
,
…
,
u
,
u
.
Similarly we have
(17)
f
u
=
g
u
=
T
u
,
u
,
…
,
u
,
u
.
Since
(
f
,
S
)
and
(
g
,
T
)
are jointly
2
k
-weakly compatible pairs, we have
(18)
f
z
=
f
f
u
=
f
S
u
,
u
,
…
,
u
,
u
=
S
f
u
,
f
u
,
…
,
f
u
,
f
u
=
S
z
,
z
,
…
,
z
,
z
,
(19)
g
z
=
T
z
,
z
,
…
,
z
,
z
(20)
d
f
z
,
z
=
d
f
z
,
g
u
=
d
S
z
,
z
,
…
,
z
,
z
,
T
u
,
u
,
…
,
u
,
u
from
18
,
17
≤
λ
max
d
g
z
,
f
u
,
d
f
z
,
g
u
,
d
g
z
,
f
u
,
d
f
z
,
g
u
,
…
,
d
g
z
,
f
u
,
d
f
z
,
g
u
=
λ
max
d
g
z
,
z
,
d
f
z
,
z
from
14
.
Similarly, we have
(21)
d
g
z
,
z
≤
λ
max
d
z
,
f
z
,
d
z
,
g
z
.
From (20) and (21) we have
(22)
max
d
z
,
f
z
,
d
z
,
g
z
≤
λ
max
d
z
,
f
z
,
d
z
,
g
z
which in turn yields that
(23)
f
z
=
z
,
g
z
=
z
.
From (18), (19), and (23), we have
(24)
z
=
f
z
=
g
z
=
S
z
,
z
,
…
,
z
,
z
=
T
z
,
z
,
…
,
z
,
z
.
Suppose there exists
z
′
∈
X
such that
(25)
z
′
=
f
z
′
=
g
z
′
=
S
z
′
,
z
′
,
…
,
z
′
,
z
′
=
T
z
′
,
z
′
,
…
,
z
′
,
z
′
.
Then from (2.1.2) we have
(26)
d
z
,
z
′
=
d
S
z
,
z
,
…
,
z
,
z
,
T
z
′
,
z
′
,
…
,
z
′
,
z
′
≤
λ
max
d
g
z
,
f
z
′
,
d
f
z
,
g
z
′
,
…
,
d
g
z
,
f
z
′
,
d
f
z
,
g
z
′
=
λ
d
z
,
z
′
.
This implies that
z
′
=
z
.
Thus
z
is the unique point in
X
satisfying (24).