reserve x,x1,x2,x3 for Real;

theorem
  sinh(3*x)=3*sinh(x)+4*(sinh(x))|^3
proof
  sinh(3*x)=sinh.(x+x+x) by SIN_COS2:def 2
    .=(sinh.(2*x))*(cosh.x) + (cosh.(2*x))*(sinh.x) by SIN_COS2:21
    .=2*sinh.x*cosh.x*(cosh.x) + (cosh.(2*x))*(sinh.x) by SIN_COS2:23
    .=2*sinh.x*(cosh.x)^2+(2*(cosh.x)^2-1)*(sinh.x) by SIN_COS2:23
    .=2*sinh.x*(1+(sinh.x)^2)+(2*(cosh.x)^2-1)*(sinh.x) by Lm2
    .=2*sinh.x*(1+(sinh.x)^2)+(2*(1+(sinh.x)^2)-1)*(sinh.x) by Lm2
    .=2*sinh.x+(2+2)*sinh.x*(sinh.x)^2+sinh.x
    .=2*sinh.x+4*(sinh.x)|^1*(sinh.x)^2+sinh.x
    .=3*sinh.x+4*((sinh.x)|^1*sinh.x*sinh.x)
    .=3*sinh.x+4*((sinh.x)|^(1+1)*sinh.x) by NEWTON:6
    .=3*sinh.x+4*(sinh.x)|^(2+1) by NEWTON:6
    .=3*sinh(x)+4*(sinh.x)|^3 by SIN_COS2:def 2;
  hence thesis by SIN_COS2:def 2;
end;
