Theorem
If
and
are continuous functions and
is homotopic to
- written
thenis homotopic to
Proof
Since
there is a homotopy
with
Define
by
Then
and
Since
is continuous, so is
and
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Theorem
Ifandare continuous functions andis homotopic to- writtenthenis homotopic to
Proof
Sincethere is a homotopywith
Defineby
Thenand
Sinceis continuous, so isand