 reserve x, y, y1, y2 for set;
 reserve V for Z_Module;
 reserve u, v, w for Vector of V;
 reserve F, G, H, I for FinSequence of V;
 reserve W, W1, W2, W3 for Submodule of V;
 reserve KL1, KL2 for Linear_Combination of V;
 reserve 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 Z_Module;
    A1: B9 is Submodule of V9 by ZMODUL01:42;
    assume A2: A = B;
    now
      let x be object;
      assume x in the carrier of Lin(A);
      then consider L being Linear_Combination of A such that
      A3: x = Sum(L) by STRUCT_0:def 5,ZMODUL02:64;
      Carrier(L) c= A by VECTSP_6:def 4;
      then consider K being Linear_Combination of W such that
      A4: Carrier(K) = Carrier(L) and
      A5: Sum(K) = Sum(L) by A2,Th13,XBOOLE_1:1;
      reconsider K as Linear_Combination of B by A2,A4,VECTSP_6:def 4;
      x = Sum(K) by A3,A5;
      hence x in the carrier of Lin(B) by STRUCT_0:def 5,ZMODUL02:64;
    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 consider K being Linear_Combination of B such that
      A7: x = Sum(K) by STRUCT_0:def 5,ZMODUL02:64;
      consider L being Linear_Combination of V such that
      A8: Carrier(L) = Carrier(K) and
      A9: Sum(L) = Sum(K) by Th12;
      reconsider L as Linear_Combination of A by A2,A8,VECTSP_6:def 4;
      x = Sum(L) by A7,A9;
      hence x in the carrier of Lin(A) by STRUCT_0:def 5,ZMODUL02:64;
    end;
    then the carrier of Lin(B) c= the carrier of Lin(A);
    hence thesis by A1,A6,XBOOLE_0:def 10,ZMODUL01:45;
  end;
