If we have proved the additive property for vectors at right-angles, then we have it for all pairs:
P(a+b) | = | P(a^{perp}+kb + b) for some constant k, where a^{perp} is orthogonal to a |
= | P(a^{perp} + (k+1)b) | |
= | P(a^{perp}) + (k+1)P(b) by hypothesized linearlity | |
= | P(a) + P(b) |
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