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Given the following graphs:

enter image description here

They describe the response for two different colors . In addition that both colors are metamerism.

My question is why? how can I prove it?

Thanks

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The simple physical model of the eye (or indeed a typical camera) is that it records just three values for each pixel. In your eye, this is because you have three different types of cone cells, called S, M, and L, peaking in blue, green, and red wavelengths. In a camera, the light is passed through blue, green, and red filters before having its intensity recorded.

The result is projecting the full spectral information onto a three-dimensional subspace of color. Mathematically, this is done by convolving the original spectrum with the response function of the cones/filters. Your eyes' response functions are given here:

cone response functions

Imagine writing these as functions of wavelength, e.g. $S(\lambda)$ for the S cone. Then if your source has a spectrum $F(\lambda)$, the S cone will record a response of $$ R_S = \int\limits_\text{visible light} F(\lambda) S(\lambda)\, \mathrm{d}\lambda. $$ Similarly you can calculate $R_M$ and $R_L$. Then any spectra $F_1,F_2$ for which $R_{1S} = R_{2S}$, $R_{1M} = R_{2M}$, and $R_{1L} = R_{2L}$, the colors appear the same and they are said to be metameric.

We conclude that the map $F \mapsto (R_S,R_M,R_L)$ that calculates the responses is not injective -- different $F$'s can map to the same triplet. It is also not surjective -- there are triplets of responses that cannot be generated from any $F$. See this question for more on so-called imaginary colors.

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