
theorem Th18:
  for L being right-distributive non empty doubleLoopStr for p1,
p2 being Polynomial of L for x being Element of L holds x * (p1 + p2) = (x * p1
  ) + (x * p2)
proof
  let L be right-distributive non empty doubleLoopStr, p1,p2 be Polynomial
  of L;
  let x be Element of L;
  set f = x * (p1 + p2), g = (x * p1) + (x * p2);
A1: now
    let i9 be object;
    assume i9 in dom f;
    then reconsider i = i9 as Element of NAT;
    f.i = x*(p1+p2).i by POLYNOM5:def 4
      .= x*(p1.i+p2.i) by NORMSP_1:def 2
      .= x*p1.i+x*p2.i by VECTSP_1:def 2
      .= (x*p1).i+x*p2.i by POLYNOM5:def 4
      .= (x*p1).i+(x*p2).i by POLYNOM5:def 4
      .= g.i by NORMSP_1:def 2;
    hence f.i9 = g.i9;
  end;
  dom f = NAT by FUNCT_2:def 1
    .= dom g by FUNCT_2:def 1;
  hence thesis by A1,FUNCT_1:2;
end;
