reserve V for RealLinearSpace;

theorem Th11:
  for v being VECTOR of V, X being Subspace of V st not v in X for
  y being VECTOR of X + Lin{v}, W being Subspace of X + Lin{v} st v = y & W = X
  holds X + Lin{v} is_the_direct_sum_of W,Lin{y}
proof
  let 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,Th10;
  hence the RLSStruct of X + Lin{v} = W + Lin{y} by A3,Th9;
  assume W /\ Lin{y} <> (0).(X + Lin{v});
  then consider z being VECTOR of X + Lin{v} such that
A4: not(z in W /\ Lin{y} iff z in (0).(X + Lin{v})) by RLSUB_1:31;
  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,RLSUB_2:3;
    then consider a being Real such that
A7: z = a * y by RLVECT_4:8;
A8: z in W by A5,RLSUB_2:3;
    now
      per cases;
      suppose
        a = 0;
        then z = 0.(X + Lin{v}) by A7,RLVECT_1:10;
        hence contradiction by A6,RLSUB_1:17;
      end;
      suppose
A9:     a <> 0;
        y = 1*y by RLVECT_1:def 8
          .= a"*a*y by A9,XCMPLX_0:def 7
          .= a"*(a*y) by RLVECT_1:def 7;
        hence contradiction by A1,A2,A3,A8,A7,RLSUB_1:21;
      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,RLVECT_3:29;
    hence contradiction by A10,RLSUB_1:17;
  end;
end;
