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Number-systems-arithmetics
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Proof
(related to
Proposition: Uniqueness of Real One
)
By the
existence of real one
, we have
x=x\cdot 1
all
x\in\mathbb R\quad ( * ).
Suppose,
1^{\ast}
is any (other)
real number
, for which
x=x\cdot 1^{\ast}
all
x\in\mathbb R\quad ( * * ).
By
( * )
, we have
1^{\ast}=1^{\ast}\cdot 1.
Applying the
commutativity law for multiplying real numbers
, we get
1^{\ast}=1\cdot 1^{\ast}.
By
( * * )
we get
1^{\ast}=1.
Thus, the
real number one
1
is unique.
∎
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References
Bibliography
Forster Otto
: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
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