The statement is false, {v1+v2,v2+v3,v3+v1} is not a set of mutually orthogonal vectors.
Proof
(v1+v2)⋅(v2+v3)=v1⋅v2+v1⋅v3+v2⋅v2+v2⋅v3
Since { v1, v2, v3} is a set of mutually orthogonal vectors,
v1⋅v2=v2⋅v3=v1⋅v3=0(v1+v2)⋅(v2+v3)=v2⋅v2=∥v2∥2
Alternatively
(v1+v2)⋅(v3+v1)=∥v1∥2(v2+v3)⋅(v3+v1)=∥v3∥2
The dot products of the set of vectors are not zero. Hence, they are not mutually orthogonal.
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