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How to prove a statement about vector products?

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a1×b=a1×c => a1×(b-c)=0

=> a1=0 or b-c=0 => a1=λ(b-c) for some scalar λ

a2×b=a2×c => a2×(b-c)=0

=> a2=0 or b-c=0 => a2=λ(b-c) for some scalar λ

And I don't know what to do next?
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Counterexample: b=0, and c is some nonzero vector perpendicular to both a1 and a2.

Edit: added 'nonzero vector'

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