
theorem hcon2:
for R being Ring
for a being Element of R holds LC(a|R) = a
proof
let R be Ring; let a be Element of R;
H: 0 - 1 < 0 & 1 - 1 >= 0;
per cases;
suppose A: a = 0.R;
  then B: a|R = 0_.(R) by RING_4:13;
  then len(a|R) = 0 by POLYNOM4:3;
  then len(a|R) -' 1 = 0 by H,XREAL_0:def 2;
  then LC(a|R) = (a|R).0 by RATFUNC1:def 6;
  hence thesis by A,B;
  end;
suppose a <> 0.R;
  then A: deg(a|R) = 0 by RING_4:21;
  deg(a|R) = len(a|R) - 1 by HURWITZ:def 2;
  then len(a|R) -' 1 = 0 by A,XREAL_0:def 2;
  then LC(a|R) = (a|R).0 by RATFUNC1:def 6;
  hence thesis by poly0;
  end;
end;
