 reserve RNS1,RNS2 for RealLinearSpace;

theorem Th12:
the RLSStruct of RNS1 = the RLSStruct of RNS2
implies
  for FFr be FinSequence of RNS1,
      fr be Function of RNS1,REAL,
      Fv be FinSequence of RNS2,
      fv be Function of RNS2,REAL st fr = fv & FFr = Fv
  holds fr(#)FFr = fv(#)Fv
proof
assume A1: the RLSStruct of RNS1 = the RLSStruct of RNS2;
let FFr be FinSequence of RNS1,
    fr be Function of RNS1,REAL,
    Fv be FinSequence of RNS2,
    fv be Function of RNS2,REAL;
assume A2: fr = fv & FFr = Fv; then
A3: len(fv(#)Fv) = len FFr by RLVECT_2:def 7;
for i being Nat st i in dom (fv(#)Fv) holds
  (fv(#)Fv).i = (fr.(FFr/.i)) * (FFr/.i)
proof
  let i be Nat;
  assume i in dom (fv(#)Fv);
  hence (fv(#)Fv).i = (fv.(Fv/.i)) * (Fv/.i) by RLVECT_2:def 7
                  .= (fr.(FFr/.i)) * (FFr/.i) by A2,A1;
end;
hence thesis by A3,A1,RLVECT_2:def 7;
end;
