reserve V for non empty RLSStruct;
reserve x,y,y1 for set;
reserve v for VECTOR of V;
reserve a,b for Real;
reserve V for non empty addLoopStr;
reserve F for FinSequence-like PartFunc of NAT,V;
reserve f,f9,g for sequence of V;
reserve v,u for Element of V;
reserve j,k,n for Nat;
reserve V for RealLinearSpace;
reserve v for VECTOR of V;
reserve F,G,H,I for FinSequence of V;
reserve V for add-associative right_zeroed right_complementable non empty
  addLoopStr;
reserve F for FinSequence of V;
reserve v,v1,v2,u,w for Element of V;
reserve j,k for Nat;

theorem
  for V being RealLinearSpace, a being Real, v,u being VECTOR of V holds
  a * Sum<* v,u *> = a * v + a * u
proof
  let V be RealLinearSpace, a be Real, v,u be VECTOR of V;
  thus a * Sum<* v,u *> = a * (v + u) by Th45
    .= a * v + a * u by Def5;
end;
