let K be Field; :: thesis: for V being VectSp of K
for v being Vector of V
for X being Subspace of V
for y being Vector of (X + ())
for W being Subspace of X + () st v = y & X = W & not v in X holds
for w being Vector of (X + ()) ex x being Vector of X ex r being Element of K st w |-- (W,()) = [x,(r * v)]

let V be VectSp of K; :: thesis: for v being Vector of V
for X being Subspace of V
for y being Vector of (X + ())
for W being Subspace of X + () st v = y & X = W & not v in X holds
for w being Vector of (X + ()) ex x being Vector of X ex r being Element of K st w |-- (W,()) = [x,(r * v)]

let v be Vector of V; :: thesis: for X being Subspace of V
for y being Vector of (X + ())
for W being Subspace of X + () st v = y & X = W & not v in X holds
for w being Vector of (X + ()) ex x being Vector of X ex r being Element of K st w |-- (W,()) = [x,(r * v)]

let X be Subspace of V; :: thesis: for y being Vector of (X + ())
for W being Subspace of X + () st v = y & X = W & not v in X holds
for w being Vector of (X + ()) ex x being Vector of X ex r being Element of K st w |-- (W,()) = [x,(r * v)]

let y be Vector of (X + ()); :: thesis: for W being Subspace of X + () st v = y & X = W & not v in X holds
for w being Vector of (X + ()) ex x being Vector of X ex r being Element of K st w |-- (W,()) = [x,(r * v)]

let W be Subspace of X + (); :: thesis: ( v = y & X = W & not v in X implies for w being Vector of (X + ()) ex x being Vector of X ex r being Element of K st w |-- (W,()) = [x,(r * v)] )
assume that
A1: v = y and
A2: X = W and
A3: not v in X ; :: thesis: for w being Vector of (X + ()) ex x being Vector of X ex r being Element of K st w |-- (W,()) = [x,(r * v)]
let w be Vector of (X + ()); :: thesis: ex x being Vector of X ex r being Element of K st w |-- (W,()) = [x,(r * v)]
consider v1, v2 being Vector of (X + ()) such that
A4: w |-- (W,()) = [v1,v2] by Th17;
A5: X + () is_the_direct_sum_of W, Lin {y} by A1, A2, A3, Th14;
then v1 in W by ;
then reconsider x = v1 as Vector of X by A2;
v2 in Lin {y} by A5, A4, Th7;
then consider r being Element of K such that
A6: v2 = r * y by Th3;
take x ; :: thesis: ex r being Element of K st w |-- (W,()) = [x,(r * v)]
take r ; :: thesis: w |-- (W,()) = [x,(r * v)]
thus w |-- (W,()) = [x,(r * v)] by ; :: thesis: verum