 reserve RNS1,RNS2 for RealLinearSpace;

theorem Th16:
the RLSStruct of RNS1 = the RLSStruct of RNS2
implies
  for Ar be Subset of RNS2,
      At be Subset of RNS1 st Ar = At
  holds [#]Lin Ar = [#]Lin 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;
hereby
  let x be object;
  assume x in [#] (Lin Ar); then
  x in Lin Ar; then
  consider L being Linear_Combination of Ar such that
  A3: x = Sum L by RLVECT_3:14;
  reconsider L1 = L as Linear_Combination of RNS1 by Th7,A1;
  (Carrier L1 = Carrier L & Carrier L c= Ar) by RLVECT_2:def 6; then
  A4: L1 is Linear_Combination of At by A2, RLVECT_2:def 6;
  Sum L1 = Sum L by Th14, A1; then
  x in Lin At by A3, A4, RLVECT_3:14;
  hence x in [#] (Lin At);
end;
let x be object;
assume x in [#] (Lin At); then
x in Lin At; then
consider L being Linear_Combination of At such that
A5: x = Sum L by RLVECT_3:14;
reconsider L1 = L as Linear_Combination of RNS2 by Th7,A1;
(Carrier L1 = Carrier L & Carrier L c= At) by RLVECT_2:def 6; then
A6: L1 is Linear_Combination of Ar by A2, RLVECT_2:def 6;
Sum L1 = Sum L by Th14,A1; then
x in Lin Ar by A5, A6, RLVECT_3:14;
hence x in [#] (Lin Ar);
end;
