 reserve RNS1,RNS2 for RealLinearSpace;

theorem Th21:
for RNS1,RNS2 be RealLinearSpace
   st the RLSStruct of RNS1 = the RLSStruct of RNS2
    & RNS1 is finite-dimensional
holds RNS2 is finite-dimensional & dim RNS2 = dim RNS1
proof
let RNS1,RNS2 be RealLinearSpace;
assume A1: the RLSStruct of RNS1 = the RLSStruct of RNS2
         & RNS1 is finite-dimensional; then
consider A being finite Subset of RNS1 such that
A2: A is Basis of RNS1 by RLVECT_5:def 1;
A3: dim RNS1 = card A by RLVECT_5:def 2, A2, A1;
reconsider B = A as finite Subset of RNS2 by A1;
A4: B is Basis of RNS2 by Th20, A1, A2;
hence RNS2 is finite-dimensional by RLVECT_5:def 1;
hence dim RNS2 = dim RNS1 by RLVECT_5:def 2, A3, A4;
end;
