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

theorem
  cosh(2*y)+cos(2*z) = 2+2*((sinh y)^2-(sin z)^2) & cosh(2*y)-cos(2*z) =
  2*((sinh y)^2+(sin z)^2)
proof
A1: cosh(2*y)-cos(2*z) = 1+2*(sinh y)^2-cos(2*z) by Th27
    .= 1+2*(sinh y)^2-(1-2*(sin z)^2) by SIN_COS5:7
    .= 2*((sinh y)^2+(sin z)^2);
  cosh(2*y)+cos(2*z) = 1+2*(sinh y)^2+cos(2*z) by Th27
    .= 1+2*(sinh y)^2+(1-2*(sin z)^2) by SIN_COS5:7
    .= 2+2*((sinh y)^2-(sin z)^2);
  hence thesis by A1;
end;
