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

theorem
  (sinh(y-z)+sinh(y)+sinh(y+z))/(cosh(y-z)+cosh(y)+cosh(y+z)) = tanh(y)
proof
  (sinh(y-z)+sinh(y)+sinh(y+z))/(cosh(y-z)+cosh(y)+cosh(y+z)) = (sinh(y)*
  cosh(z)-cosh(y)*sinh(z)+sinh(y)+sinh(y+z)) /(cosh(y-z)+cosh(y)+cosh(y+z)) by
Lm10
    .= (sinh(y)+(sinh(y)*cosh(z)-cosh(y)*sinh(z)) +(sinh(y)*cosh(z)+cosh(y)*
  sinh(z)))/(cosh(y)+cosh(y-z)+cosh(y+z)) by Lm10
    .= sinh(y)*(1+(cosh(z)+cosh(z))) / (cosh(y)+(cosh(y)*cosh(z)-sinh(y)*
  sinh(z))+cosh(y+z)) by Lm10
    .= sinh(y)*(1+(cosh(z)+cosh(z))) /(cosh(y)+(cosh(y)*cosh(z)-sinh(y)*sinh
  (z)) +(cosh(y)*cosh(z)+sinh(y)*sinh(z))) by Lm10
    .= sinh(y)*(1+(cosh(z)+cosh(z)))/(cosh(y)*(1+(cosh(z)+cosh(z))))
    .= sinh(y)/cosh(y) * ((1+(cosh(z)+cosh(z)))/(1+(cosh(z)+cosh(z)))) by
XCMPLX_1:76
    .= sinh(y)/cosh(y)*1 by Lm12,XCMPLX_1:60
    .= tanh(y) by Th1;
  hence thesis;
end;
