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
for w being VECTOR of (X + (Lin {v})) ex x being VECTOR of X ex r being Real st w |-- W,(Lin {y}) = [x,(r * v)]
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
for w being VECTOR of (X + (Lin {v})) ex x being VECTOR of X ex r being Real st w |-- W,(Lin {y}) = [x,(r * v)]
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
for w being VECTOR of (X + (Lin {v})) ex x being VECTOR of X ex r being Real st w |-- W,(Lin {y}) = [x,(r * v)]
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
for w being VECTOR of (X + (Lin {v})) ex x being VECTOR of X ex r being Real st w |-- W,(Lin {y}) = [x,(r * v)]
let W be Subspace of X + (Lin {v}); :: thesis: ( v = y & X = W & not v in X implies for w being VECTOR of (X + (Lin {v})) ex x being VECTOR of X ex r being Real st w |-- W,(Lin {y}) = [x,(r * v)] )
assume A1: ( v = y & X = W & not v in X ) ; :: thesis: for w being VECTOR of (X + (Lin {v})) ex x being VECTOR of X ex r being Real st w |-- W,(Lin {y}) = [x,(r * v)]
let w be VECTOR of (X + (Lin {v})); :: thesis: ex x being VECTOR of X ex r being Real st w |-- W,(Lin {y}) = [x,(r * v)]
A2: X + (Lin {v}) is_the_direct_sum_of W, Lin {y} by A1, Th24;
consider v1, v2 being VECTOR of (X + (Lin {v})) such that
A3: w |-- W,(Lin {y}) = [v1,v2] by Th27;
v1 in W by A2, A3, Th17;
then reconsider x = v1 as VECTOR of X by A1, STRUCT_0:def 5;
v2 in Lin {y} by A2, A3, Th17;
then consider r being Real such that
A4: v2 = r * y by RLVECT_4:11;
take x ; :: thesis: ex r being Real st w |-- W,(Lin {y}) = [x,(r * v)]
take r ; :: thesis: w |-- W,(Lin {y}) = [x,(r * v)]
thus w |-- W,(Lin {y}) = [x,(r * v)] by A1, A3, A4, RLSUB_1:22; :: thesis: verum