let V be finite-dimensional RealUnitarySpace; :: thesis: dim V = dim ((Omega). V)

consider I being finite Subset of V such that

A1: I is Basis of V by Def1;

A2: (Omega). V = UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) by RUSUB_1:def 3

.= Lin I by A1, RUSUB_3:def 2 ;

( card I = dim V & I is linearly-independent ) by A1, Def2, RUSUB_3:def 2;

hence dim V = dim ((Omega). V) by A2, Th9; :: thesis: verum

consider I being finite Subset of V such that

A1: I is Basis of V by Def1;

A2: (Omega). V = UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) by RUSUB_1:def 3

.= Lin I by A1, RUSUB_3:def 2 ;

( card I = dim V & I is linearly-independent ) by A1, Def2, RUSUB_3:def 2;

hence dim V = dim ((Omega). V) by A2, Th9; :: thesis: verum