reserve x, y, z, r, s, t for Real;

theorem
  sgn x = sgn (1/x)
proof
A1: 0 < x implies sgn x = sgn (1/x)
    proof
      assume
A2:   0 < x;
      sgn (1/x) = 1/(sgn x) by Th21;
      then sgn (1/x) = 1/1 by A2,Def2
        .= 1;
      hence thesis by A2,Def2;
    end;
    x < 0 implies sgn x = sgn (1/x)
    proof
      assume
A3:   x < 0;
      then sgn x = -1 by Def2;
      then sgn (1/x) = 1/(-1) by Th21;
      hence thesis by A3,Def2;
    end;
    hence thesis by A1;
end;
