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;

theorem Th14:
  x in Lin(A) iff ex l st x = Sum(l)
proof
  thus x in Lin(A) implies ex l st x = Sum(l)
  proof
    assume x in Lin(A);
    then x in the carrier of Lin(A) by STRUCT_0:def 5;
    then x in the set of all Sum(l)  by Def2;
    hence thesis;
  end;
  given k being Linear_Combination of A such that
A1: x = Sum(k);
  x in the set of all Sum(l) by A1;
  then x in the carrier of Lin(A) by Def2;
  hence thesis by STRUCT_0:def 5;
end;
