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

theorem
  (sinh(5*x)+2*sinh(3*x)+sinh x)/(sinh(7*x)+2*sinh(5*x)+sinh(3*x)) =
  sinh(3*x)/sinh(5*x)
proof
  (sinh(5*x)+2*sinh(3*x)+sinh(x))/(sinh(7*x)+2*sinh(5*x)+sinh(3*x)) = (
  sinh(3*x+2*x)+2*sinh(3*x)+sinh(x)) /(sinh((5+2)*x)+2*sinh(5*x)+sinh(3*x))
    .= ((sinh(3*x)*cosh(2*x)+cosh(3*x)*sinh(2*x))+2*sinh(3*x) +sinh(x))/(
  sinh((5+2)*x)+2*sinh(5*x)+sinh(3*x)) by Lm10
    .= (sinh(3*x)*(cosh(2*x)+2)+(cosh(2*x+1*x)*sinh(2*x)+sinh(x))) /(sinh(5*
  x+2*x)+2*sinh(5*x)+sinh(3*x))
    .= (sinh(3*x)*(cosh(2*x)+2)+((cosh(2*x)*cosh(x) +sinh(2*x)*sinh(x))*sinh
  (2*x)+sinh(x))) /(sinh(5*x+2*x)+2*sinh(5*x)+sinh(3*x)) by Lm10
    .= (sinh(3*x)*(cosh(2*x)+2)+(cosh(2*x)*cosh(x)*sinh(2*x) +((sinh(2*x))^2
  +1)*sinh(x))) /(sinh(5*x+2*x)+2*sinh(5*x)+sinh(3*x))
    .= (sinh(3*x)*(cosh(2*x)+2)+(cosh(2*x)*cosh(x)*sinh(2*x) +(((cosh(2*x))
^2-(sinh(2*x))^2)+(sinh(2*x))^2)*sinh(x))) /(sinh(5*x+2*x)+2*sinh(5*x)+sinh(3*x
  )) by Lm3
    .= (sinh(3*x)*(cosh(2*x)+2)+(cosh(2*x)*(sinh(2*x)*cosh(x) +cosh(2*x)*
  sinh(x))))/(sinh(5*x+2*x)+2*sinh(5*x)+sinh(3*x))
    .= (sinh(3*x)*(cosh(2*x)+2)+(cosh(2*x)*sinh(2*x+1*x))) /(sinh(5*x+2*x)+2
  *sinh(5*x)+sinh(3*x)) by Lm10
    .= sinh(3*x)*((cosh(2*x)+2)+cosh(2*x))/((sinh(5*x)*cosh(2*x) +cosh(5*x)*
  sinh(2*x))+sinh(5*x)*2+sinh(3*x)) by Lm10
    .= sinh(3*x)*((cosh(2*x)+2)+cosh(2*x))/(sinh(5*x)*(cosh(2*x)+2) +(cosh(3
  *x+2*x)*sinh(2*x)+sinh(3*x)))
    .= sinh(3*x)*((cosh(2*x)+2)+cosh(2*x))/(sinh(5*x)*(cosh(2*x)+2) +((cosh(
  3*x)*cosh(2*x)+sinh(3*x)*sinh(2*x)) *sinh(2*x)+sinh(3*x))) by Lm10
    .= sinh(3*x)*((cosh(2*x)+2)+cosh(2*x))/(sinh(5*x)*(cosh(2*x)+2) +(cosh(3
  *x)*cosh(2*x)*sinh(2*x)+sinh(3*x)*((sinh(2*x))^2+1)))
    .= sinh(3*x)*((cosh(2*x)+2)+cosh(2*x))/(sinh(5*x)*(cosh(2*x)+2) +(cosh(3
*x)*cosh(2*x)*sinh(2*x)+sinh(3*x)*(((cosh(2*x))^2 -(sinh(2*x))^2)+(sinh(2*x))^2
  ))) by Lm3
    .= sinh(3*x)*((cosh(2*x)+2)+cosh(2*x))/(sinh(5*x)*(cosh(2*x)+2) +cosh(2*
  x)*(sinh(2*x)*cosh(3*x)+cosh(2*x)*sinh(3*x)))
    .= sinh(3*x)*((cosh(2*x)+2)+cosh(2*x))/(sinh(5*x)*(cosh(2*x)+2) +cosh(2*
  x)*sinh(2*x+3*x)) by Lm10
    .= sinh(3*x)*((cosh(2*x)+2)+cosh(2*x)) /(sinh(5*x)*((cosh(2*x)+2)+cosh(2
  *x)))
    .= sinh(3*x)/sinh(5*x) by Lm13,XCMPLX_1:91;
  hence thesis;
end;
