theorem :: PRVECT_3:20

for X, Y being RealNormSpace-Sequence ex I being Function of (product <*(product X),(product Y)*>),(product (X ^ Y)) st
( I is one-to-one & I is onto & ( for x being Point of (product X)
for y being Point of (product Y) ex x1, y1 being FinSequence st
( x = x1 & y = y1 & I . <*x,y*> = x1 ^ y1 ) ) & ( for v, w being Point of (product <*(product X),(product Y)*>) holds I . (v + w) = (I . v) + (I . w) ) & ( for v being Point of (product <*(product X),(product Y)*>)
for r being Element of REAL holds I . (r * v) = r * (I . v) ) & I . (0. (product <*(product X),(product Y)*>)) = 0. (product (X ^ Y)) & ( for v being Point of (product <*(product X),(product Y)*>) holds ||.(I . v).|| = ||.v.|| ) )