
theorem Th1:
  for V being RealUnitarySpace, A being Subset of V, x be object holds
  x in Lin(A) iff ex l being Linear_Combination of A st x = Sum(l)
proof
  let V be RealUnitarySpace;
  let A be Subset of V;
  let x be object;
  thus x in Lin(A) implies ex l being Linear_Combination of A 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) where l is Linear_Combination of A
    by Def1;
    hence thesis;
  end;
  given k being Linear_Combination of A such that
A1: x = Sum(k);
  x in the set of all Sum(l) where l is Linear_Combination of A  by A1;
  then x in the carrier of Lin(A) by Def1;
  hence thesis by STRUCT_0:def 5;
end;
