let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; :: thesis: for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF holds {(0. V)} is linearly-closed

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: {(0. V)} is linearly-closed

thus for v, u being Element of V st v in {(0. V)} & u in {(0. V)} holds

v + u in {(0. V)} :: according to VECTSP_4:def 1 :: thesis: for a being Element of GF

for v being Element of V st v in {(0. V)} holds

a * v in {(0. V)}

a * v in {(0. V)}

let v be Element of V; :: thesis: ( v in {(0. V)} implies a * v in {(0. V)} )

assume A1: v in {(0. V)} ; :: thesis: a * v in {(0. V)}

then v = 0. V by TARSKI:def 1;

hence a * v in {(0. V)} by A1, VECTSP_1:14; :: thesis: verum

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: {(0. V)} is linearly-closed

thus for v, u being Element of V st v in {(0. V)} & u in {(0. V)} holds

v + u in {(0. V)} :: according to VECTSP_4:def 1 :: thesis: for a being Element of GF

for v being Element of V st v in {(0. V)} holds

a * v in {(0. V)}

proof

let a be Element of GF; :: thesis: for v being Element of V st v in {(0. V)} holds
let v, u be Element of V; :: thesis: ( v in {(0. V)} & u in {(0. V)} implies v + u in {(0. V)} )

assume ( v in {(0. V)} & u in {(0. V)} ) ; :: thesis: v + u in {(0. V)}

then ( v = 0. V & u = 0. V ) by TARSKI:def 1;

then v + u = 0. V by RLVECT_1:4;

hence v + u in {(0. V)} by TARSKI:def 1; :: thesis: verum

end;assume ( v in {(0. V)} & u in {(0. V)} ) ; :: thesis: v + u in {(0. V)}

then ( v = 0. V & u = 0. V ) by TARSKI:def 1;

then v + u = 0. V by RLVECT_1:4;

hence v + u in {(0. V)} by TARSKI:def 1; :: thesis: verum

a * v in {(0. V)}

let v be Element of V; :: thesis: ( v in {(0. V)} implies a * v in {(0. V)} )

assume A1: v in {(0. V)} ; :: thesis: a * v in {(0. V)}

then v = 0. V by TARSKI:def 1;

hence a * v in {(0. V)} by A1, VECTSP_1:14; :: thesis: verum