reserve x,y for object, X,Y,Z for set;
reserve a,b for Real;
reserve k for Element of NAT;
reserve V for RealLinearSpace;
reserve W1,W2,W3 for Subspace of V;
reserve v,v1,v2,u for VECTOR of V;
reserve A,B,C for Subset of V;
reserve T for finite Subset of V;
reserve L,L1,L2 for Linear_Combination of V;
reserve l for Linear_Combination of A;
reserve F,G,H for FinSequence of the carrier of V;
reserve f,g for Function of the carrier of V, REAL;
reserve p,q,r for FinSequence;
reserve M for non empty set;
reserve CF for Choice_Function of M;
reserve l0 for Linear_Combination of {}(the carrier of V);

theorem Th18:
  for W being strict Subspace of V holds A = the carrier of W
  implies Lin(A) = W
proof
  let W be strict Subspace of V;
  assume
A1: A = the carrier of W;
  now
    let v;
    thus v in Lin(A) implies v in W
    proof
      assume v in Lin(A);
      then
A2:   ex l st v = Sum(l) by Th14;
      A is linearly-closed by A1,RLSUB_1:34;
      then v in the carrier of W by A1,A2,RLVECT_2:29;
      hence thesis by STRUCT_0:def 5;
    end;
    v in W iff v in the carrier of W by STRUCT_0:def 5;
    hence v in W implies v in Lin(A) by A1,Th15;
  end;
  hence thesis by RLSUB_1:31;
end;
