a1×b=a1×c => a1×(b-c)=0

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

a2×b=a2×c => a2×(b-c)=0

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

If a1,a2 are non-zero, then b-c is either zero (=> b=c) or co-linear with both a1 and a2. But if a1,a2 aren't co-linear (with each other), b-c can't be co-linear with both.