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

theorem
  (sinh(y)+sinh(z))/(sinh(y)-sinh(z)) = tanh((y+z)/2)*coth((y-z)/2)
proof
  (sinh(y)+sinh(z))/(sinh(y)-sinh(z)) = 2*sinh(y/2+z/2)*cosh(y/2-z/2)/(
  sinh(y)-sinh(z)) by Lm11
    .= 2*sinh(y/2+z/2)*cosh(y/2-z/2) / (2*sinh(y/2-z/2)*cosh(y/2+z/2)) by Lm11
    .= 2*sinh(y/2+z/2)*cosh(y/2-z/2) / (2*cosh(y/2+z/2)*sinh(y/2-z/2))
    .= 2*sinh(y/2+z/2)/(2*cosh(y/2+z/2)) * (cosh(y/2-z/2)/sinh(y/2-z/2)) by
XCMPLX_1:76
    .= 2*sinh(y/2+z/2)/(2*cosh(y/2+z/2)) * coth(y/2-z/2) by SIN_COS5:def 1
    .= 2/2*(sinh(y/2+z/2)/cosh(y/2+z/2)) * coth(y/2-z/2) by XCMPLX_1:76
    .= tanh((y+z)/2) * coth((y-z)/2) by Th1;
  hence thesis;
end;
