reserve x, y, z, w for Real;
reserve n for Element of NAT;

theorem
  x >= 0 implies tanh x >= 0
proof
A1: cosh(x) >= 1 by Lm1;
  assume
A2: x >= 0;
  per cases by A2;
  suppose
    x > 0;
    then sinh(x) >= 0 by SIN_COS5:46;
    then sinh(x)/cosh(x) >= 0 by A1;
    hence thesis by Th1;
  end;
  suppose
    x = 0;
    hence thesis by Th1;
  end;
end;
