
theorem Th7:
  for L be Abelian add-associative right_zeroed
  right_complementable distributive non empty doubleLoopStr for a,b being
  Element of L, p being sequence of L holds (a+b)*p = a*p + b*p
proof
  let L be Abelian add-associative right_zeroed right_complementable
  distributive non empty doubleLoopStr;
  let a,b be Element of L, p be sequence of L;
  for i be Element of NAT holds ((a+b)*p).i = (a*p + b*p).i
  proof
    let i be Element of NAT;
    thus ((a+b)*p).i = (a+b)*p.i by POLYNOM5:def 4
      .= a*p.i + b*p.i by VECTSP_1:def 7
      .= (a*p).i + b*p.i by POLYNOM5:def 4
      .= (a*p).i + (b*p).i by POLYNOM5:def 4
      .= (a*p + b*p).i by NORMSP_1:def 2;
  end;
  hence thesis by FUNCT_2:63;
end;
