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

theorem
  (cosh(y)+cosh(z))/(cosh(y)-cosh(z)) = coth((y+z)/2)*coth((y-z)/2)
proof
  (cosh(y)+cosh(z))/(cosh(y)-cosh(z)) = 2*cosh(y/2+z/2)*cosh(y/2-z/2)/(
  cosh(y)-cosh(z)) by Lm11
    .= 2*cosh(y/2+z/2)*cosh(y/2-z/2)/(2*sinh(y/2-z/2)*sinh((y+z)/2)) by Lm11
    .= 2*cosh(y/2+z/2)*cosh(y/2-z/2)/(2*sinh(y/2+z/2)*sinh((y-z)/2))
    .= 2*cosh(y/2+z/2)/(2*sinh(y/2+z/2)) * (cosh(y/2-z/2)/sinh(y/2-z/2)) by
XCMPLX_1:76
    .= 2*cosh(y/2+z/2)/(2*sinh(y/2+z/2))*coth(y/2-z/2) by SIN_COS5:def 1
    .= 2/2*(cosh(y/2+z/2)/sinh(y/2+z/2))*coth(y/2-z/2) by XCMPLX_1:76
    .= coth((y+z)/2)*coth((y-z)/2) by SIN_COS5:def 1;
  hence thesis;
end;
