
theorem Th6:
  for L be Abelian add-associative right_zeroed
right_complementable distributive non empty doubleLoopStr for a being Element
  of L, p,q being sequence of L holds a*(p+q)=a*p + a*q
proof
  let L be Abelian add-associative right_zeroed right_complementable
  distributive non empty doubleLoopStr;
  let a be Element of L, p,q be sequence of L;
  for i be Element of NAT holds (a*(p+q)).i = (a*p + a*q).i
  proof
    let i be Element of NAT;
    a*((p+q).i) = a*(p.i + q.i) by NORMSP_1:def 2
      .= a*(p.i) + a*(q.i) by VECTSP_1:def 7
      .= (a*p).i + a*(q.i) by POLYNOM5:def 4
      .= (a*p).i + (a*q).i by POLYNOM5:def 4
      .= (a*p + a*q).i by NORMSP_1:def 2;
    hence thesis by POLYNOM5:def 4;
  end;
  hence thesis by FUNCT_2:63;
end;
