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

theorem
  sinh(y)*sinh(z)*sinh(z-y) + sinh(z)*sinh(w)*sinh(w-z) + sinh(w)*sinh(y
  )*sinh(y-w) + (sinh(z-y)*sinh(w-z)*sinh(y-w)) = 0
proof
  sinh(y)*sinh(z)*sinh(z-y) + sinh(z)*sinh(w)*sinh(w-z) + sinh(w)*sinh(y)*
sinh(y-w) + (sinh(z-y)*sinh(w-z)*sinh(y-w)) = (1/2*(cosh(y+z)-cosh(y-z)))*sinh(
z-y) + sinh(z)*sinh(w)*sinh(w-z) + sinh(w)*sinh(y)*sinh(y-w) + sinh(z-y)*sinh(w
  -z)*sinh(y-w) by Th11
    .= 1/2*(cosh(y+z)*sinh(z-y)-cosh(y-z)*sinh(z-y)) + sinh(z)*sinh(w)*sinh(
  w-z) + sinh(w)*sinh(y)*sinh(y-w) + sinh(z-y)*sinh(w-z)*sinh(y-w)
    .= 1/2*(1/2*(sinh(y+z+(z-y))-sinh(y+z-(z-y)))-cosh(y-z)*sinh(z-y)) +
  sinh(z)*sinh(w)*sinh(w-z) + sinh(w)*sinh(y)*sinh(y-w) + sinh(z-y)*sinh(w-z)*
  sinh(y-w) by Th11
    .= 1/2*(1/2*(sinh(2*z)-sinh(y+y))-1/2*(sinh(y-z+-(y-z))-sinh(y-z-(z-y)))
  ) +sinh(z)*sinh(w)*sinh(w-z) +sinh(w)*sinh(y)*sinh(y-w) +sinh(z-y)*sinh(w-z)*
  sinh(y-w) by Th11
    .= 1/2*(1/2*(sinh(2*z)-sinh(2*y))-1/2*(sinh.0-sinh(2*(y-z)))) + sinh(z)*
  sinh(w)*sinh(w-z) + sinh(w)*sinh(y)*sinh(y-w) + sinh(z-y)*sinh(w-z)*sinh(y-w)
  by SIN_COS2:def 2
    .= 1/2*(1/2*(sinh(2*z)-sinh(2*y)--sinh(2*(y-z)))) + 1/2*(cosh(z+w)-cosh(
z-w))*sinh(w-z) + sinh(w)*sinh(y)*sinh(y-w) + sinh(z-y)*sinh(w-z)*sinh(y-w) by
Th11,SIN_COS2:16
    .= 1/2*(1/2*(sinh(2*z)-sinh(2*y)--sinh(2*(y-z))) + (cosh(z+w)*sinh(w-z)-
cosh(z-w)*sinh(w-z))) + sinh(w)*sinh(y)*sinh(y-w) + sinh(z-y)*sinh(w-z)*sinh(y-
  w)
    .= 1/2*(1/2*(sinh(2*z)-sinh(2*y)--sinh(2*(y-z))) + (1/2*(sinh(z+w+(w-z))
-sinh(z+w-(w-z))) - cosh(z-w)*sinh(w-z))) + sinh(w)*sinh(y)*sinh(y-w) + sinh(z-
  y)*sinh(w-z)*sinh(y-w) by Th11
    .= 1/2*(1/2*(sinh(2*z)-sinh(2*y)--sinh(2*(y-z))) + (1/2*(sinh(2*w)-sinh(
  2*z)) - 1/2*(sinh(z-w+(w-z))-sinh(z-w-(w-z))))) + sinh(w)*sinh(y)*sinh(y-w) +
  sinh(z-y)*sinh(w-z)*sinh(y-w) by Th11
    .= 1/2*(1/2*(sinh(2*z)-sinh(2*y)--sinh(2*(y-z))) + (1/2*(sinh(2*w)-sinh(
2*z)) - 1/2*(0-sinh(z-w-(w-z))))) + sinh(w)*sinh(y)*sinh(y-w) + sinh(z-y)*sinh(
  w-z)*sinh(y-w) by SIN_COS2:16,def 2
    .= 1/2*(1/2*(sinh(2*z)-sinh(2*y)--sinh(2*(y-z)) + (sinh(2*w)-sinh(2*z)--
  sinh(2*(z-w))))) + 1/2*(cosh(w+y)-cosh(w-y))*sinh(y-w) + sinh(z-y)*sinh(w-z)*
  sinh(y-w) by Th11
    .= 1/2*(1/2*(sinh(2*z)-sinh(2*y)--sinh(2*(y-z)) + (sinh(2*w)-sinh(2*z)--
sinh(2*(z-w)))) + (cosh(w+y)*sinh(y-w)-cosh(w-y)*sinh(y-w))) + sinh(z-y)*sinh(w
  -z)*sinh(y-w)
    .= 1/2*(1/2*(sinh(2*z)-sinh(2*y)--sinh(2*(y-z)) + (sinh(2*w)-sinh(2*z)--
sinh(2*(z-w)))) + (1/2*(sinh(w+y+(y-w))-sinh(w+y-(y-w)))-cosh(w-y)*sinh(y-w)))
  + sinh(z-y)*sinh(w-z)*sinh(y-w) by Th11
    .= 1/2*(1/2*(sinh(2*z)-sinh(2*y)--sinh(2*(y-z)) + (sinh(2*w)-sinh(2*z)--
sinh(2*(z-w)))) + (1/2*(sinh(2*y)-sinh(2*w))-1/2*(sinh(w-y +-(w-y))-sinh(w-y--(
  w-y))))) + sinh(z-y)*sinh(w-z)*sinh(y-w) by Th11
    .= 1/2*(1/2*(sinh(2*z)-sinh(2*y)--sinh(2*(y-z)) + (sinh(2*w)-sinh(2*z)--
sinh(2*(z-w)))) + (1/2*(sinh(2*y)-sinh(2*w))-1/2*(0-sinh(2*(w-y))))) + sinh(z-y
  )*sinh(w-z)*sinh(y-w) by SIN_COS2:16,def 2
    .= 1/2*(1/2*(sinh(2*(y-z))+sinh(2*(z-w))+sinh(2*(w-y)))) + 1/2*(cosh(z-y
  +-(z-w)) - cosh(z-y-(w-z)))*sinh(y-w) by Th11
    .= 1/2*(1/2*(sinh(2*(y-z))+sinh(2*(z-w))+sinh(2*(w-y))) + (cosh(w-y)*
  sinh(y-w)-cosh(z-y-(w-z))*sinh(y-w)))
    .= 1/2*(1/2*(sinh(2*(y-z))+sinh(2*(z-w))+sinh(2*(w-y))) + (1/2*(sinh(w-y
  +-(w-y))-sinh(w-y-(y-w)))-cosh(z-y-(w-z))*sinh(y-w))) by Th11
    .= 1/2*(1/2*(sinh(2*(y-z))+sinh(2*(z-w))+sinh(2*(w-y))) + (1/2*(0-sinh(w
  -y-(y-w)))-cosh(z-y-(w-z))*sinh(y-w))) by SIN_COS2:16,def 2
    .= 1/2*(1/2*(sinh(2*(y-z))+sinh(2*(z-w))+sinh(2*(w-y))) + (1/2*-sinh(2*(
  w-y))-1/2*(sinh(z-y+(z-w)+(y-w))-sinh(z-y+(z-w)-(y-w))))) by Th11
    .= 1/2*(1/2*(sinh(2*(y-z))+sinh(2*(z-w))+sinh(2*(w-y))) + 1/2*((sinh(-(2
  *(y-z)))-sinh(2*(z-w)))+-sinh(2*(w-y))))
    .= 1/2*(1/2*(sinh(2*(y-z))+sinh(2*(z-w))+sinh(2*(w-y))) + 1/2*(-sinh(2*(
  y-z))-sinh(2*(z-w))+-sinh(2*(w-y)))) by Lm8;
  hence thesis;
end;
