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

theorem
  (sinh y)^2-(cosh z)^2 = sinh(y+z)*sinh(y-z)-1
proof
  (sinh y)^2-(cosh z)^2 = (sinh y)^2-(cosh z)^2*1
    .= (sinh y)^2-(cosh z)^2*((cosh y)^2-(sinh y)^2) by Lm3
    .= (sinh y)^2+((cosh z)^2*(sinh y)^2 -(cosh y)^2*((cosh z)^2-(sinh z)^2+
  (sinh z)^2))
    .= (sinh y)^2+((cosh z)^2*(sinh y)^2 -(cosh y)^2*(1+(sinh z)^2)) by Lm3
    .= (sinh y)^2+((sinh(y)*cosh(z)+cosh(y)*sinh(z)) *(sinh(y)*cosh(z)-cosh(
  y)*sinh z)-(cosh y)^2)
    .= (sinh y)^2+(sinh(y+z) *(sinh(y)*cosh(z)-cosh(y)*sinh(z))-(cosh y)^2)
  by Lm10
    .= sinh(y+z)*sinh(y-z)-(cosh y)^2+(sinh y)^2 by Lm10
    .= sinh(y+z)*sinh(y-z)-((cosh y)^2-(sinh y)^2)
    .= sinh(y+z)*sinh(y-z)-1 by Lm3;
  hence thesis;
end;
