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

for V1 being Subset of V st V1 is linearly-closed holds

for v, u being Element of V st v in V1 & u in V1 holds

v - u in V1

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for V1 being Subset of V st V1 is linearly-closed holds

for v, u being Element of V st v in V1 & u in V1 holds

v - u in V1

let V1 be Subset of V; :: thesis: ( V1 is linearly-closed implies for v, u being Element of V st v in V1 & u in V1 holds

v - u in V1 )

assume A1: V1 is linearly-closed ; :: thesis: for v, u being Element of V st v in V1 & u in V1 holds

v - u in V1

let v, u be Element of V; :: thesis: ( v in V1 & u in V1 implies v - u in V1 )

assume that

A2: v in V1 and

A3: u in V1 ; :: thesis: v - u in V1

- u in V1 by A1, A3, Th2;

hence v - u in V1 by A1, A2; :: thesis: verum

for V1 being Subset of V st V1 is linearly-closed holds

for v, u being Element of V st v in V1 & u in V1 holds

v - u in V1

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for V1 being Subset of V st V1 is linearly-closed holds

for v, u being Element of V st v in V1 & u in V1 holds

v - u in V1

let V1 be Subset of V; :: thesis: ( V1 is linearly-closed implies for v, u being Element of V st v in V1 & u in V1 holds

v - u in V1 )

assume A1: V1 is linearly-closed ; :: thesis: for v, u being Element of V st v in V1 & u in V1 holds

v - u in V1

let v, u be Element of V; :: thesis: ( v in V1 & u in V1 implies v - u in V1 )

assume that

A2: v in V1 and

A3: u in V1 ; :: thesis: v - u in V1

- u in V1 by A1, A3, Th2;

hence v - u in V1 by A1, A2; :: thesis: verum