let GF be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; :: thesis: for V being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr of 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 VectSp-like Abelian add-associative right_zeroed VectSpStr of 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 ( Carrier (a * L) c= Carrier L & Carrier L c= A ) by Def7, Th58;
then Carrier (a * L) c= A by XBOOLE_1:1;
hence a * L is Linear_Combination of A by Def7; :: thesis: verum