x\cdot y=1 \} $ the object of convertible elements in $R$ and define an apartness relation on a cartesian product $R^n$ as it follows: let $x,y \in R^n$, then $x \# y$ iff $\exists i (x_i - y_i \in InvR)$.

We assume that $R$ is a local formally real Pythagorean Archimedean ring and field of quotients [@Ko].

Having the apartness relation on $R^n$ we can develop the theory of intuitionistic linear algebra so as it was made by Heyting in [@He]. In particular, the intuitionistic theory of linear equations is valid. The notion of basis of $R$-module are also defined.

Having the apparat of linear algebra and using the assumptions with respect to $R$ that were given above we define a scalar product on $R^n$ and show that the main metrical properties of $R^n$ do not differ from classical one. For example, if the determinant of matrix of a scalar product is aparted of zero than the matrix is invertible.

For $R$-modules $U$ and $V$ we define a tensor product $U\otimes V$ and all operations of tensor algebra in a standard manner. Since $R^n$ is a $R$-module we can define the notions of covariant and contrvariant tensors on $R^n$, and 