let V be RealLinearSpace; :: thesis: for v being VECTOR of V
for X being Subspace of V
for y being VECTOR of (X + (Lin {v}))
for W being Subspace of X + (Lin {v}) st v = y & X = W & not v in X holds
y |-- (W,(Lin {y})) = [(0. W),y]
let v be VECTOR of V; :: thesis: for X being Subspace of V
for y being VECTOR of (X + (Lin {v}))
for W being Subspace of X + (Lin {v}) st v = y & X = W & not v in X holds
y |-- (W,(Lin {y})) = [(0. W),y]
let X be Subspace of V; :: thesis: for y being VECTOR of (X + (Lin {v}))
for W being Subspace of X + (Lin {v}) st v = y & X = W & not v in X holds
y |-- (W,(Lin {y})) = [(0. W),y]
let y be VECTOR of (X + (Lin {v})); :: thesis: for W being Subspace of X + (Lin {v}) st v = y & X = W & not v in X holds
y |-- (W,(Lin {y})) = [(0. W),y]
let W be Subspace of X + (Lin {v}); :: thesis: ( v = y & X = W & not v in X implies y |-- (W,(Lin {y})) = [(0. W),y] )
assume ( v = y & X = W & not v in X ) ; :: thesis: y |-- (W,(Lin {y})) = [(0. W),y]
then X + (Lin {v}) is_the_direct_sum_of W, Lin {y} by Th11;
then y |-- (W,(Lin {y})) = [(0. (X + (Lin {v}))),y] by Th7, RLVECT_4:9;
hence y |-- (W,(Lin {y})) = [(0. W),y] by RLSUB_1:11; :: thesis: verum