reserve x,y,t for Real;

theorem
  cosh(5*x)=5*cosh(x)-20*(cosh(x))|^3+16*(cosh(x))|^5
proof
  set t = cosh.x, u = sinh.x;
  cosh(5*x) = cosh.(4*x+x) by SIN_COS2:def 4
    .=(cosh.(2*(2*x)))*t+(sinh.(4*x))*u by SIN_COS2:20
    .=(1+2*(sinh.(2*x))^2)*t+(sinh.(4*x))*u by Th69
    .=(1+2*((2*u)*t)^2)*t+(sinh.(2*(2*x)))*u by SIN_COS2:23
    .=t+8*u^2*(t^2*t)+(sinh.(2*(2*x)))*u
    .=t+8*u^2*(t|^1*t*t)+(sinh.(2*(2*x)))*u
    .=t+8*u^2*(t|^(1+1)*t)+(sinh.(2*(2*x)))*u by NEWTON:6
    .=t+8*(sinh.x)^2*(t|^(2+1))+(sinh.(2*(2*x)))*u by NEWTON:6
    .=t+8*(t^2-1)*(t|^3)+(sinh.(2*(2*x)))*u by Th71
    .=t+8*(t|^3*t*t)-8*t|^3+(sinh.(2*(2*x)))*u
    .=t+8*(t|^(3+1)*t)-8*t|^3+(sinh.(2*(2*x)))*u by NEWTON:6
    .=t+8*(t|^(4+1))-8*t|^3+(sinh.(2*(2*x)))*u by NEWTON:6
    .=t+8*(t|^5)-8*t|^3+(2*(sinh.(2*x))*(cosh.(2*x)))*u by SIN_COS2:23
    .=t+8*(t|^5)-8*t|^3+(2*(2*u*t)*(cosh.(2*x)))*u by SIN_COS2:23
    .=t+8*(t|^5)-8*t|^3+4*u^2*t*(cosh.(2*x))
    .=t+8*(t|^5)-8*t|^3+4*(t^2-1)*t*(cosh.(2*x)) by Th71
    .=t+8*(t|^5)-8*t|^3+4*(t*t*t)*(cosh.(2*x))-4*t*(cosh.(2*x))
    .=t+8*(t|^5)-8*t|^3+4*(t|^1*t*t)*(cosh.(2*x))-4*t*(cosh.(2*x))
    .=t+8*(t|^5)-8*t|^3+4*(t|^(1+1)*t)*(cosh.(2*x))-4*t*(cosh.(2*x)) by
NEWTON:6
    .=t+8*(t|^5)-8*t|^3+4*(t|^(2+1)) *(cosh.(2*x))-4*t*(cosh.(2*x)) by NEWTON:6
    .=t+8*(t|^5)-8*t|^3+4*t|^3*(2*t^2-1)-4*t*(cosh.(2*x)) by SIN_COS2:23
    .=t+8*(t|^5)-8*t|^3+8*(t|^3*t*t)-4*t|^3-4*t*(cosh.(2*x))
    .=t+8*(t|^5)-8*t|^3+8*(t|^(3+1)*t)-4*t|^3-4*t*(cosh.(2*x)) by NEWTON:6
    .=t+8*(t|^5)-8*t|^3+8*(t|^(4+1))-4*t|^3-4*t*(cosh.(2*x)) by NEWTON:6
    .=t+16*(t|^5)-12*t|^3-4*t*(2*t^2-1) by SIN_COS2:23
    .=5*t+16*(t|^5)-12*t|^3-8*(t*t*t)
    .=5*t+16*(t|^5)-12*t|^3-8*(t|^1*t*t)
    .=5*t+16*(t|^5)-12*t|^3-8*(t|^(1+1)*t) by NEWTON:6
    .=5*t+16*(t|^5)-12*t|^3-8*(t|^(2+1)) by NEWTON:6
    .=5*t-20*t|^3+16*(t|^5)
    .=5*(cosh x)-20*t|^3+16*(t|^5) by SIN_COS2:def 4
    .=5*(cosh x)-20*(cosh x)|^3+16*(t|^5) by SIN_COS2:def 4
    .=5*cosh(x)-20*(cosh x)|^3+16*(cosh x)|^5 by SIN_COS2:def 4;
  hence thesis;
end;
