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forall x in C', y in C': cartesian(x *C' y) = cartesian(x) *C cartesian(y).Take arbitrary x in C' and y in C'. We then have
(*) holds because of the definition "shift" and for every x in R,
cartesian(x *C' y) = cartesian(x0y0, shift(x1+y1)) = x0y0cos(shift(x1+y1)) + (x0y0sin(shift(x1+y1)))i = (*) x0y0cos(x1+y1) + (x0y0sin(x1+y1))i.
sin(x+2 pi ) = sin(x), cos(x+2 pi ) = cos(x)