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additive group basis of 313 bers bijective called Cauchy's criterion circular arc closed and bounded closed real segment closed segment coefficient commutative group commutative law consisting continuous map convex cosets defined denote the set distance preserving maps distinct points elements of 313 finite set follows greatest lower bound group homomorphism group isomorphism group operation Hence identity element injective inner product integer least upper bound linear combination linear endomorphism linearly independent subset map that sends ment neutral element noncollinear points nonvoid nonzero element notion numerical measure obtain one-dimensional subspaces open set open side ordered pairs orthogonal linear map pair p,q parallel phism plane points of 3l2 positive real number progression of points pure quaternion rational numbers real vector space ring rotation satisfies scalar multiple Section sequence of terms side of L(p,q square patches Suppose surjective tion two-dimensional subspace uniformly continuous unit circle unit disk verifies write