let GF be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; :: thesis: for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF
for V1 being Subset of V st the carrier of V = V1 holds
V1 is linearly-closed
let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF; :: thesis: for V1 being Subset of V st the carrier of V = V1 holds
V1 is linearly-closed
let V1 be Subset of V; :: thesis: ( the carrier of V = V1 implies V1 is linearly-closed )
assume A1: the carrier of V = V1 ; :: thesis: V1 is linearly-closed
hence for v, u being Element of V st v in V1 & u in V1 holds
v + u in V1 ; :: according to VECTSP_4:def 1 :: thesis: for a being Element of GF
for v being Element of V st v in V1 holds
a * v in V1
let a be Element of GF; :: thesis: for v being Element of V st v in V1 holds
a * v in V1
let v be Element of V; :: thesis: ( v in V1 implies a * v in V1 )
assume v in V1 ; :: thesis: a * v in V1
thus a * v in V1 by A1; :: thesis: verum