 reserve RNS1,RNS2 for RealLinearSpace;

theorem
the RLSStruct of RNS1 = the RLSStruct of RNS2
implies
for X being object
  holds
 (X is Subspace of RNS2
    iff
  X is Subspace of RNS1)
proof
assume A1: the RLSStruct of RNS1 = the RLSStruct of RNS2;
let X be object;
hereby
  assume X is Subspace of RNS2; then
  reconsider V = X as Subspace of RNS2;
  A2: the carrier of V c= the carrier of RNS2
    & 0.V = 0.(RNS2)
    & the addF of V = (the addF of RNS2) || the carrier of V
    & the Mult of V = (the Mult of RNS2) | [:REAL, the carrier of V:]
      by RLSUB_1:def 2;
  0.V = 0.(RNS1) by A2,A1;
  hence X is Subspace of RNS1 by A1, A2, RLSUB_1:def 2;
end;
assume X is Subspace of RNS1; then
reconsider V = X as Subspace of RNS1;
A4: the carrier of V c= the carrier of RNS1
  & 0.V = 0.(RNS1)
  & the addF of V = (the addF of RNS1) || the carrier of V
  & the Mult of V = (the Mult of RNS1) | [:REAL, the carrier of V:]
    by RLSUB_1:def 2;
0.V = 0.(RNS2) by A4,A1;
hence X is Subspace of RNS2 by A1,A4,RLSUB_1:def 2;
end;
