let V be non empty CLSStruct ; :: thesis: for A being Subset of V
for a being Complex
for L being C_Linear_Combination of V st L is C_Linear_Combination of A holds
a * L is C_Linear_Combination of A
let A be Subset of V; :: thesis: for a being Complex
for L being C_Linear_Combination of V st L is C_Linear_Combination of A holds
a * L is C_Linear_Combination of A
let a be Complex; :: thesis: for L being C_Linear_Combination of V st L is C_Linear_Combination of A holds
a * L is C_Linear_Combination of A
let L be C_Linear_Combination of V; :: thesis: ( L is C_Linear_Combination of A implies a * L is C_Linear_Combination of A )
assume A1: L is C_Linear_Combination of A ; :: thesis: a * L is C_Linear_Combination of A

A2: now :: thesis: ( a <> 0c implies a * L is C_Linear_Combination of A )

assume a <> 0c ; :: thesis: a * L is C_Linear_Combination of A
then Carrier (a * L) = Carrier L by Th24;
hence a * L is C_Linear_Combination of A by A1, Def4; :: thesis: verum

end;

( a = 0c implies a * L = ZeroCLC V ) by Th25;
hence a * L is C_Linear_Combination of A by A2, Th4; :: thesis: verum