begin
Lm1:
for GF being non empty right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over GF
for a, b being Element of GF
for v being Element of V holds (a - b) * v = (a * v) - (b * v)
Lm2:
for GF being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over GF
for W being Subspace of V
for V1 being Subset of V st the carrier of W = V1 holds
V1 is linearly-closed
Lm3:
for GF being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over GF
for W being Subspace of V holds (0. V) + W = the carrier of W
Lm4:
for GF being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr over GF
for v being Element of V
for W being Subspace of V holds
( v in W iff v + W = the carrier of W )
:: Auxiliary theorems.
::