 reserve RNS1,RNS2 for RealLinearSpace;

theorem
the RLSStruct of RNS1 = the RLSStruct of RNS2
implies
  for L be Linear_Combination of RNS2,
      S be Linear_Combination of RNS1
st L = S
  holds
  Sum L = Sum S
proof
assume A1:the RLSStruct of RNS1 = the RLSStruct of RNS2;
let L be Linear_Combination of RNS2,
    S be Linear_Combination of RNS1;
assume A2: L = S;
consider F being FinSequence of RNS2 such that
A3: F is one-to-one & rng F = Carrier L
  & Sum L = Sum (L (#) F) by RLVECT_2:def 8;
reconsider E = F as FinSequence of RNS1 by A1;
A4: Sum(L(#)F) = Sum(S(#)E) by A2,A1,Th12,Th13;
reconsider SS=Sum L as Element of RNS1 by A1;
thus thesis by RLVECT_2:def 8,A2,A3,A4;
end;
