Partial Differentiation of Real Binary Functions

For simplicity, we adopt the following convention: x, x0, y, y0, r are real numbers, z, z0 are elements of R2, Z is a subset of R2, f , f1, f2 are partial functions from R2 to R, R is a rest, and L is a linear function. Next we state two propositions: (1) domproj(1, 2) = R2 and rng proj(1, 2) = R and for all elements x, y of R holds (proj(1, 2))(〈x, y〉) = x. (2) domproj(2, 2) = R2 and rng proj(2, 2) = R and for all elements x, y of R holds (proj(2, 2))(〈x, y〉) = y.