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 Element of GF

for A being Subset of V

for L being Linear_Combination of V st L is Linear_Combination of A holds

a * L 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 Element of GF

for A being Subset of V

for L being Linear_Combination of V st L is Linear_Combination of A holds

a * L is Linear_Combination of A

let a be Element of GF; :: thesis: for A being Subset of V

for L being Linear_Combination of V st L is Linear_Combination of A holds

a * L is Linear_Combination of A

let A be Subset of V; :: thesis: for L being Linear_Combination of V st L is Linear_Combination of A holds

a * L is Linear_Combination of A

let L be Linear_Combination of V; :: thesis: ( L is Linear_Combination of A implies a * L is Linear_Combination of A )

assume L is Linear_Combination of A ; :: thesis: a * L is Linear_Combination of A

then A1: Carrier L c= A by Def4;

Carrier (a * L) c= Carrier L by Th28;

then Carrier (a * L) c= A by A1;

hence a * L is Linear_Combination of A by Def4; :: thesis: verum

for a being Element of GF

for A being Subset of V

for L being Linear_Combination of V st L is Linear_Combination of A holds

a * L 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 Element of GF

for A being Subset of V

for L being Linear_Combination of V st L is Linear_Combination of A holds

a * L is Linear_Combination of A

let a be Element of GF; :: thesis: for A being Subset of V

for L being Linear_Combination of V st L is Linear_Combination of A holds

a * L is Linear_Combination of A

let A be Subset of V; :: thesis: for L being Linear_Combination of V st L is Linear_Combination of A holds

a * L is Linear_Combination of A

let L be Linear_Combination of V; :: thesis: ( L is Linear_Combination of A implies a * L is Linear_Combination of A )

assume L is Linear_Combination of A ; :: thesis: a * L is Linear_Combination of A

then A1: Carrier L c= A by Def4;

Carrier (a * L) c= Carrier L by Th28;

then Carrier (a * L) c= A by A1;

hence a * L is Linear_Combination of A by Def4; :: thesis: verum