
theorem LM86:
  for x be Real holds |.x.| = (sgn x) * x
proof
  let x be Real;
A1: 0 < x implies |.x.| = (sgn x) * x
  proof
    assume A2: 0 < x; then
    (sgn x) * x = 1 * x by ABSVALUE:def 2
               .= x;
    hence thesis by A2,ABSVALUE:def 1;
  end;
A4: x < 0 implies |.x.| = (sgn x) * x
  proof
    assume A5: x < 0; then
    (sgn x) * x = (-1) * x by ABSVALUE:def 2
               .= -x;
    hence thesis by A5,ABSVALUE:def 1;
  end;
  x = 0 implies |.x.| = (sgn x) * x;
  hence thesis by A1,A4;
end;
