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, B being Subset of V

for l being Linear_Combination of A st A c= B holds

l is Linear_Combination of B

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, B being Subset of V

for l being Linear_Combination of A st A c= B holds

l is Linear_Combination of B

let A, B be Subset of V; :: thesis: for l being Linear_Combination of A st A c= B holds

l is Linear_Combination of B

let l be Linear_Combination of A; :: thesis: ( A c= B implies l is Linear_Combination of B )

assume A1: A c= B ; :: thesis: l is Linear_Combination of B

Carrier l c= A by Def4;

then Carrier l c= B by A1;

hence l is Linear_Combination of B by Def4; :: thesis: verum

for A, B being Subset of V

for l being Linear_Combination of A st A c= B holds

l is Linear_Combination of B

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, B being Subset of V

for l being Linear_Combination of A st A c= B holds

l is Linear_Combination of B

let A, B be Subset of V; :: thesis: for l being Linear_Combination of A st A c= B holds

l is Linear_Combination of B

let l be Linear_Combination of A; :: thesis: ( A c= B implies l is Linear_Combination of B )

assume A1: A c= B ; :: thesis: l is Linear_Combination of B

Carrier l c= A by Def4;

then Carrier l c= B by A1;

hence l is Linear_Combination of B by Def4; :: thesis: verum