reserve r,t for Real;
reserve i for Integer;
reserve k,n for Nat;
reserve p for Polynomial of F_Real;
reserve e for Element of F_Real;
reserve L for non empty ZeroStr;
reserve z,z0,z1,z2 for Element of L;

theorem Th44:
for L being add-associative right_zeroed right_complementable
            distributive associative well-unital domRing-like
            non degenerated doubleLoopStr
for p,q being Polynomial of L holds LC(p *' q) = LC(p) * LC(q)
proof
let L be add-associative right_zeroed right_complementable
         distributive associative well-unital domRing-like
         non degenerated doubleLoopStr;
let p,q be Polynomial of L;
per cases;
suppose
AS: p <> 0_.(L) & q <> 0_.(L);
len(p*'q) = deg(p*'q) + 1
              .= deg(p) + deg(q) + 1 by AS,HURWITZ:23
              .= len(p) - 1 + (len(q) - 1) + 1;
hence LC(p *' q) = (p*'q).(len p + len q - 1 -'1)
                .= LC(p) * LC(q) by AS,lemmul;
end;
    suppose
A30:  p is zero;
      then p*'q=0_.L by Th17;
      hence LC(p*'q) = 0.L * LC q by FUNCOP_1:7
      .= LC p * LC q by A30,FUNCOP_1:7;
    end;
    suppose
A31:  q is zero;
      then p*'q=0_.L by POLYNOM3:34;
      hence LC(p*'q) = LC p * 0.L by FUNCOP_1:7
      .= LC p * LC q by A31,FUNCOP_1:7;
    end;
end;
