reserve V for RealLinearSpace,
  W for Subspace of V,
  x, y, y1, y2 for set,
  i, n for Element of NAT,
  v for VECTOR of V,
  KL1, KL2 for Linear_Combination of V,
  X for Subset of V;

theorem Th20:
  for A being Subset of V, B being Subset of W st A = B holds Lin( A) = Lin(B)
proof
  let A be Subset of V, B be Subset of W;
  reconsider B9= Lin(B), V9= V as RealLinearSpace;
A1: B9 is Subspace of V9 by RLSUB_1:27;
  assume
A2: A = B;
  now
    let x be object;
    assume x in the carrier of Lin(A);
    then x in Lin(A) by STRUCT_0:def 5;
    then consider L being Linear_Combination of A such that
A3: x = Sum(L) by RLVECT_3:14;
    Carrier(L) c= A by RLVECT_2:def 6;
    then consider K being Linear_Combination of W such that
A4: Carrier(K) = Carrier(L) and
A5: Sum(K) = Sum(L) by A2,Th12,XBOOLE_1:1;
    reconsider K as Linear_Combination of B by A2,A4,RLVECT_2:def 6;
    x = Sum(K) by A3,A5;
    then x in Lin(B) by RLVECT_3:14;
    hence x in the carrier of Lin(B) by STRUCT_0:def 5;
  end;
  then
A6: the carrier of Lin(A) c= the carrier of Lin(B);
  now
    let x be object;
    assume x in the carrier of Lin(B);
    then x in Lin(B) by STRUCT_0:def 5;
    then consider K being Linear_Combination of B such that
A7: x = Sum(K) by RLVECT_3:14;
    consider L being Linear_Combination of V such that
A8: Carrier(L) = Carrier(K) and
A9: Sum(L) = Sum(K) by Th11;
    reconsider L as Linear_Combination of A by A2,A8,RLVECT_2:def 6;
    x = Sum(L) by A7,A9;
    then x in Lin(A) by RLVECT_3:14;
    hence x in the carrier of Lin(A) by STRUCT_0:def 5;
  end;
  then the carrier of Lin(B) c= the carrier of Lin(A);
  then the carrier of Lin(A) = the carrier of Lin(B) by A6,XBOOLE_0:def 10;
  hence thesis by A1,RLSUB_1:30;
end;
