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

theorem
  cosh(2*y)+cosh(2*z)+cosh(2*w)+cosh(2*(y+z+w)) = 4*cosh(z+w)*cosh(w+y)*
  cosh(y+z)
proof
  cosh(2*y)+cosh(2*z)+cosh(2*w)+cosh(2*(y+z+w)) = 2*(1/2*(cosh(2*(y+z+w))+
  cosh(2*y))+1/2*(cosh(2*w)+cosh(2*z)))
    .= 2*(1/2*(2*cosh(2*(y+z+w)/2+2*y/2)*cosh(2*(y+z+w)/2-2*y/2)) + 1/2*(
  cosh(2*w)+cosh(2*z))) by Lm11
    .= 2*((cosh(z+w)*cosh((y+z+w)+y)) + 1/2*(2*cosh((2*w)/2+2*z/2)*cosh(2*w/
  2-2*z/2))) by Lm11
    .= 2*(cosh(z+w)*(cosh(2*y+(z+w))+cosh(w-z)))
    .= 2*(cosh(z+w) *(2*cosh((2*y+(z+w))/2+(w-z)/2)*cosh((2*y+(z+w))/2-(w-z)
  /2))) by Lm11
    .= 4*cosh(z+w)*cosh(w+y)*cosh(y+z);
  hence thesis;
end;
