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 A being Subset of V

for L1, L2 being Linear_Combination of V st L1 is Linear_Combination of A & L2 is Linear_Combination of A holds

L1 - L2 is Linear_Combination of A

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 A being Subset of V

for L1, L2 being Linear_Combination of V st L1 is Linear_Combination of A & L2 is Linear_Combination of A holds

L1 - L2 is Linear_Combination of A

let A be Subset of V; :: thesis: for L1, L2 being Linear_Combination of V st L1 is Linear_Combination of A & L2 is Linear_Combination of A holds

L1 - L2 is Linear_Combination of A

let L1, L2 be Linear_Combination of V; :: thesis: ( L1 is Linear_Combination of A & L2 is Linear_Combination of A implies L1 - L2 is Linear_Combination of A )

assume that

A1: L1 is Linear_Combination of A and

A2: L2 is Linear_Combination of A ; :: thesis: L1 - L2 is Linear_Combination of A

- L2 is Linear_Combination of A by A2, Th31;

hence L1 - L2 is Linear_Combination of A by A1, Th24; :: thesis: verum

for A being Subset of V

for L1, L2 being Linear_Combination of V st L1 is Linear_Combination of A & L2 is Linear_Combination of A holds

L1 - L2 is Linear_Combination of A

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 A being Subset of V

for L1, L2 being Linear_Combination of V st L1 is Linear_Combination of A & L2 is Linear_Combination of A holds

L1 - L2 is Linear_Combination of A

let A be Subset of V; :: thesis: for L1, L2 being Linear_Combination of V st L1 is Linear_Combination of A & L2 is Linear_Combination of A holds

L1 - L2 is Linear_Combination of A

let L1, L2 be Linear_Combination of V; :: thesis: ( L1 is Linear_Combination of A & L2 is Linear_Combination of A implies L1 - L2 is Linear_Combination of A )

assume that

A1: L1 is Linear_Combination of A and

A2: L2 is Linear_Combination of A ; :: thesis: L1 - L2 is Linear_Combination of A

- L2 is Linear_Combination of A by A2, Th31;

hence L1 - L2 is Linear_Combination of A by A1, Th24; :: thesis: verum