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

theorem
  (sinh(y)+sinh(z))/(cosh(y)+cosh(z)) = tanh(y/2+z/2) & (sinh(y)-sinh(z)
  )/(cosh(y)+cosh(z)) = tanh(y/2-z/2)
proof
A1: cosh(y/2-z/2) <> 0 by Lm1;
A2: cosh(y/2+z/2) <> 0 by Lm1;
A3: (sinh(y)-sinh(z))/(cosh(y)+cosh(z)) = 2*sinh(y/2-z/2)*cosh(y/2+z/2)/(
  cosh(y)+cosh(z)) by Lm11
    .= 2*(sinh(y/2-z/2)*cosh(y/2+z/2))/(2*cosh(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)*cosh(y/2-z/2)))
    .= cosh(y/2+z/2)*sinh(y/2-z/2)/(cosh(y/2+z/2)*cosh(y/2-z/2)) by XCMPLX_1:91
    .= cosh(y/2+z/2)/cosh(y/2+z/2)*(sinh(y/2-z/2)/cosh(y/2-z/2)) by XCMPLX_1:76
    .= 1*(sinh(y/2-z/2)/cosh(y/2-z/2)) by A2,XCMPLX_1:60
    .= tanh(y/2-z/2) by Th1;
  (sinh(y)+sinh(z))/(cosh(y)+cosh(z)) = 2*sinh(y/2+z/2)*cosh(y/2-z/2)/(
  cosh(y)+cosh(z)) by Lm11
    .= 2*sinh(y/2+z/2)*cosh(y/2-z/2)/(2*cosh(y/2+z/2)*cosh(y/2-z/2)) by Lm11
    .= 2*sinh(y/2+z/2)/(2*cosh(y/2+z/2))*(cosh(y/2-z/2)/cosh(y/2-z/2)) by
XCMPLX_1:76
    .= 2*sinh(y/2+z/2)/(2*cosh(y/2+z/2))*1 by A1,XCMPLX_1:60
    .= 2/2*(sinh(y/2+z/2)/cosh(y/2+z/2)) by XCMPLX_1:76
    .= tanh(y/2+z/2) by Th1;
  hence thesis by A3;
end;
