
theorem Th18:
for L being add-associative right_zeroed right_complementable
            well-unital commutative associative distributive
            almost_left_invertible non empty doubleLoopStr
for p being Polynomial of L
for a being Element of L holds LC(a * p) = a * LC(p)
proof
let L be add-associative right_zeroed right_complementable
         well-unital commutative associative distributive
         almost_left_invertible non empty doubleLoopStr;
let p be Polynomial of L;
let a be Element of L;
per cases;
suppose A1: a = 0.L;
  then A2: a * LC(p) = 0.L;
  a * p = 0_.(L) by A1,POLYNOM5:26;
  hence thesis by A2,FUNCOP_1:7;
  end;
suppose A3: a <> 0.L;
  thus LC(a * p) = a * (p.(len(a*p)-'1)) by POLYNOM5:def 4
                .= a * LC(p) by A3,POLYNOM5:25;
  end;
end;
