 reserve RNS1,RNS2 for RealLinearSpace;

theorem Th14:
the RLSStruct of RNS1 = the RLSStruct of RNS2
implies
  for Lr be Linear_Combination of RNS2,
      Lt be Linear_Combination of RNS1 st Lr = Lt
  holds Sum Lr = Sum Lt
proof
assume A1: the RLSStruct of RNS1 = the RLSStruct of RNS2;
  let Lr be Linear_Combination of RNS2,
      Lt be Linear_Combination of RNS1;
  assume A2: Lr = Lt;
set R = RNS2;
set T = RNS1;
consider Ft being FinSequence of the carrier of RNS1 such that
A3: Ft is one-to-one & rng Ft = Carrier Lt and
A4: Sum Lt = Sum (Lt (#) Ft) by RLVECT_2:def 8;
reconsider FFr = Ft as FinSequence of the carrier of RNS2 by A1;
thus Sum Lt = Sum (Lr (#) FFr) by A2, Th12, A1, Th13, A4
           .= Sum Lr by A3, A2, RLVECT_2:def 8;
end;
