  Let F be a subfield of a commutative field extending R. Let \phi_2: F^2
\times F^2 \to F, \phi_2((x_1,x_2),(y_1,y_2))=(x_1-y_1)^2+(x_2-y_2)^2. We say
that f:R^2 \to F^2 preserves distance d \geq 0 if for each x,y \in R^2 |x-y|=d
implies \phi_2(f(x),f(y))=d^2. We prove that each unit-distance preserving
mapping f:R^2 \to F^2 has a form I \circ (\rho,\rho), where \rho: R \to F is a
field homomorphism and I: F^2 \to F^2 is an affine mapping with orthogonal
linear part.
