
theorem Th14:
  for K be Field, V be VectSp of K for v be Vector of V, X be
Subspace of V st not v in X for y be Vector of X + Lin{v}, W be Subspace of X +
  Lin{v} st v = y & W = X holds X + Lin{v} is_the_direct_sum_of W,Lin{y}
proof
  let K be Field, V be VectSp of K, v be Vector of V, X be Subspace of V such
  that
A1: not v in X;
  let y be Vector of X + Lin{v}, W be Subspace of X + Lin{v};
  assume that
A2: v = y and
A3: W = X;
  Lin{v} = Lin{y} by A2,Th13;
  hence the ModuleStr of X + Lin{v} = W + Lin{y} by A3,Th12;
  assume W /\ Lin{y} <> (0).(X + Lin{v});
  then consider z be Vector of X + Lin{v} such that
A4: not(z in W /\ Lin{y} iff z in (0).(X + Lin{v})) by VECTSP_4:30;
  per cases by A4;
  suppose that
A5: z in W /\ Lin{y} and
A6: not z in (0).(X + Lin{v});
    z in Lin{y} by A5,VECTSP_5:3;
    then consider a be Element of K such that
A7: z = a * y by Th3;
A8: z in W by A5,VECTSP_5:3;
    now
      per cases;
      suppose
        a = 0.K;
        then z = 0.(X + Lin{v}) by A7,VECTSP_1:15;
        hence contradiction by A6,VECTSP_4:17;
      end;
      suppose
A9:     a <> 0.K;
        y = (1_K)*y
          .= a"*a*y by A9,VECTSP_1:def 10
          .= a"*(a*y) by VECTSP_1:def 16;
        hence contradiction by A1,A2,A3,A8,A7,VECTSP_4:21;
      end;
    end;
    hence contradiction;
  end;
  suppose that
A10: not z in W /\ Lin{y} and
A11: z in (0).(X + Lin{v});
    z = 0.(X + Lin{v}) by A11,VECTSP_4:35;
    hence contradiction by A10,VECTSP_4:17;
  end;
end;
