reserve F for Field;
reserve VS for strict VectSp of F;
reserve u,e for set;
reserve x for set;
reserve Z1 for set;

theorem Th10:
  for x being object for VS being strict VectSp of F for S being
Subset of VS st S is non empty & S is linearly-closed holds x in Lin S implies
  x in S
proof
  let x be object, VS be strict VectSp of F, S be Subset of VS;
  assume S is non empty & S is linearly-closed;
  then consider W being strict Subspace of VS such that
A1: S = the carrier of W by VECTSP_4:34;
  assume
A2: x in Lin S;
  Lin S = W by A1,VECTSP_7:11;
  hence thesis by A2,A1,STRUCT_0:def 5;
end;
