let V be RealLinearSpace; 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; 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; 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})); 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}); ( 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 )
; 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; verum