
theorem Th57:
  for a, b, c being Ordinal st c in b
  holds b -leading_coeff (c *^ exp(b,a)) = c
proof
  let a,b,c be Ordinal;
  assume A1: c in b;
  per cases;
  suppose A2: 0 in c;
    A3: 0 in exp(b,a) by A1, ORDINAL1:14;
    thus b -leading_coeff (c *^ exp(b,a))
       = (c *^ exp(b,a)) div^ exp(b, a) by A1, A2, ORDINAL5:58
      .= (c *^ exp(b,a) +^ 0) div^ exp(b, a) by ORDINAL2:27
      .= c by A3, ORDINAL3:66;
  end;
  suppose not 0 in c;
    then A4: c = 0 by ORDINAL1:14;
    hence b -leading_coeff (c *^ exp(b,a)) = b -leading_coeff 0 by ORDINAL2:35
      .= c by A4, ORDINAL3:70;
  end;
end;
