 reserve RNS1,RNS2 for RealLinearSpace;

theorem Th17:
the RLSStruct of RNS1 = the RLSStruct of RNS2
implies
  for Ar be Subset of RNS2,
      At be Subset of RNS1 st Ar = At
  holds
    Ar is linearly-independent
  iff
    At is linearly-independent
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;
hereby
  assume A3: Ar is linearly-independent;
  now
    let L be Linear_Combination of At;
    reconsider L1 = L as Linear_Combination of RNS2 by Th7,A1;
    A4: Carrier L1 = Carrier L;
    assume Sum L = 0.(RNS1); then
    A5: 0.(RNS2) = Sum L1 by Th14,A1;
    L1 is Linear_Combination of Ar by A2, A4, RLVECT_2:def 6;
    hence Carrier L = {} by A3, A5, RLVECT_3:def 1;
  end;
  hence At is linearly-independent by RLVECT_3:def 1;
end;
assume A6: At is linearly-independent;
now
  let L be Linear_Combination of Ar;
  reconsider L1 = L as Linear_Combination of RNS1 by Th7, A1;
  Carrier L1 = Carrier L; then
  reconsider L1 = L as Linear_Combination of At by A2, RLVECT_2:def 6;
  assume Sum L = 0.(RNS2); then
  0.(RNS1) = Sum L1 by Th14, A1;
  hence Carrier L = {} by A6, RLVECT_3:def 1;
end;
hence Ar is linearly-independent by RLVECT_3:def 1;
end;
