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

theorem
  (sinh(y)-sinh z)^2-(cosh(y)-cosh z)^2 = 4*(sinh((y-z)/2))^2 & (cosh(y)
  +cosh z)^2-(sinh(y)+sinh z)^2 = 4*(cosh((y-z)/2))^2
proof
A1: (cosh(y)+cosh(z))^2-(sinh(y)+sinh(z))^2 = (2*cosh(y/2+z/2)*cosh(y/2-z/2)
  )^2-(sinh(y)+sinh(z))^2 by Lm11
    .= (2*(cosh(y/2+z/2)*cosh(y/2-z/2)))^2 - (2*sinh(y/2+z/2)*cosh(y/2-z/2))
  ^2 by Lm11
    .= 4*((cosh(y/2-z/2))^2 * ((cosh(y/2+z/2))^2 - (sinh(y/2+z/2))^2))
    .= 4*((cosh(y/2-z/2))^2*1) by Lm3
    .= 4*(cosh((y-z)/2))^2;
  (sinh(y)-sinh z)^2-(cosh(y)-cosh z)^2 = (2*sinh(y/2-z/2)*cosh(y/2+z/2))
  ^2-(cosh(y)-cosh z)^2 by Lm11
    .= (2*cosh(y/2+z/2)*sinh(y/2-z/2))^2 -(2*sinh(y/2-z/2)*sinh((y+z)/2))^2
  by Lm11
    .= 4*((sinh(y/2-z/2))^2*((cosh(y/2+z/2))^2 - (sinh(y/2+z/2))^2))
    .= 4*((sinh(y/2-z/2))^2*1) by Lm3
    .= 4*(sinh((y-z)/2))^2;
  hence thesis by A1;
end;
