 reserve RNS1,RNS2 for RealLinearSpace;

theorem
the RLSStruct of RNS1 = the RLSStruct of RNS2
implies
  for Ar be Subset of RNS2,
      At be Subset of RNS1 st Ar = At
  holds
for X be object holds
   X is Linear_Combination of Ar
     iff
   X is Linear_Combination of At
proof
assume A1: the RLSStruct of RNS1 = the RLSStruct of RNS2;
let Ar be Subset of RNS2,
    At be Subset of RNS1;
assume A2: Ar = At;
let X be object;
hereby assume
  X is Linear_Combination of Ar; then
  reconsider L=X as Linear_Combination of Ar;
  reconsider L1 = L as Linear_Combination of RNS1 by Th7,A1;
  (Carrier L1 = Carrier L & Carrier L c= Ar) by RLVECT_2:def 6;
  hence X is Linear_Combination of At by A2, RLVECT_2:def 6;
end;
assume X is Linear_Combination of At; then
  reconsider L=X as Linear_Combination of At;
  reconsider L1 = L as Linear_Combination of RNS2 by Th7, A1;
  (Carrier L1 = Carrier L & Carrier L c= At) by RLVECT_2:def 6;
  hence X is Linear_Combination of Ar by A2, RLVECT_2:def 6;
end;
