 reserve RNS1,RNS2 for RealLinearSpace;

theorem Th7:
the RLSStruct of RNS1 = the RLSStruct of RNS2
implies
  for X being set
  for n being Nat holds
    ( X is Linear_Combination of RNS2
  iff X is Linear_Combination of RNS1 )
proof
assume A1: the RLSStruct of RNS1 = the RLSStruct of RNS2;
let X be set;
let n be Nat;
hereby assume X is Linear_Combination of RNS2; then
  reconsider Lr = X as Linear_Combination of RNS2;
  consider T being finite Subset of RNS2 such that
  A2: for v being Element of RNS2 st not v in T holds
      Lr.v = 0 by RLVECT_2:def 3;
  reconsider T0=T as finite Subset of RNS1 by A1;
  thus X is Linear_Combination of RNS1 by A2,RLVECT_2:def 3,A1;
end;
assume X is Linear_Combination of RNS1; then
reconsider Lr = X as Linear_Combination of RNS1;
consider T being finite Subset of RNS1 such that
A3: for v being Element of RNS1 st not v in T holds
    Lr.v = 0 by RLVECT_2:def 3;
thus X is Linear_Combination of RNS2 by A3,RLVECT_2:def 3,A1;
end;
