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

theorem
  (sinh x)|^3 = (sinh(3*x)-3*sinh x)/4 & (sinh x)|^4 = (cosh(4*x)-4*cosh
(2*x)+3)/8 & (sinh x)|^5 = (sinh(5*x)-5*sinh(3*x)+10*sinh(x))/16 & (sinh x)|^6
= (cosh(6*x)-6*cosh(4*x)+15*cosh(2*x)-10)/32 & (sinh x)|^7 = (sinh(7*x)-7*sinh(
5*x)+21*sinh(3*x)-35*sinh(x))/64 & (sinh x)|^8 = (cosh(8*x)-8*cosh(6*x)+28*cosh
  (4*x)-56*cosh(2*x)+35)/128
proof
A1: (sinh x)|^3 = (4*(sinh(x))|^3+3*sinh(x)-3*sinh(x))/4
    .= (sinh(3*x)-3*sinh(x))/4 by SIN_COS5:43;
  then
A2: (sinh x)|^4 = (sinh(3*x)-3*sinh(x))/4*sinh(x) by POLYEQ_2:4
    .= (sinh(3*x)*sinh(x)-3*(sinh(x)*sinh(x)))/4
    .= (1/2*(cosh(3*x+1*x)-cosh(3*x-1*x))-3*(sinh(x)*sinh(x)))/4 by Th11
    .= (1/2*(cosh(4*x)-cosh(2*x))-(sinh x)^2*3)/4
    .= (1/2*(cosh(4*x)-cosh(2*x))-1/2*(cosh(2*x)-1)*3)/4 by Lm7
    .= (cosh(4*x)-4*cosh(2*x)+3)/8;
A3: (sinh x)|^5 = (sinh(x))|^(4+1)
    .= (cosh(4*x)-4*cosh(2*x)+3)/8*sinh(x) by A2,NEWTON:6
    .= (cosh(4*x)*sinh(x)-4*(cosh(2*x)*sinh(x))+3*sinh(x))/8
    .= (1/2*(sinh(4*x+1*x)-sinh(4*x-1*x))-4*(cosh(2*x)*sinh(x)) +3*sinh(x))/
  8 by Th11
    .= (1/2*(sinh(5*x)-sinh(3*x))-4*(1/2*(sinh(2*x+1*x) -sinh(2*x-1*x)))+3*
  sinh(x))/8 by Th11
    .= (sinh(5*x)-5*sinh(3*x)+10*sinh(x))/16;
A4: (sinh x)|^6 = (sinh x)|^(5+1)
    .= (sinh(5*x)-5*sinh(3*x)+10*sinh(x))/16*sinh(x) by A3,NEWTON:6
    .= (sinh(5*x)*sinh(x)-5*(sinh(3*x)*sinh(x)) +10*(sinh(x)*sinh(x)))/16
    .= (1/2*(cosh(5*x+1*x)-cosh(5*x-1*x))-5*(sinh(3*x)*sinh(x)) +10*(sinh(x)
  *sinh(x)))/16 by Th11
    .= (1/2*(cosh(6*x)-cosh(4*x))-1/2*(cosh(3*x+1*x) -cosh(3*x-1*x))*5+10*(
  sinh(x)*sinh(x)))/16 by Th11
    .= (1/2*(cosh(6*x)-6*cosh(4*x)+cosh(2*x)*5)+10*(sinh x)^2)/16
    .= (1/2*(cosh(6*x)-6*cosh(4*x)+cosh(2*x)*5)+1/2*(cosh(2*x)-1)*10)/16 by Lm7
    .= (cosh(6*x)-6*cosh(4*x)+15*cosh(2*x)-10)/32;
A5: (sinh x)|^7 = (sinh x)|^(6+1)
    .= (cosh(6*x)-6*cosh(4*x)+15*cosh(2*x)-10)/32*sinh(x) by A4,NEWTON:6
    .= ((cosh(6*x)*sinh(x)-6*cosh(4*x)*sinh(x) +15*cosh(2*x)*sinh(x)-10*sinh
  (x)))/32
    .= (1/2*(sinh(6*x+1*x)-sinh(6*x-1*x))-6*cosh(4*x)*sinh(x) +15*cosh(2*x)*
  sinh(x)-10*sinh(x))/32 by Th11
    .= (1/2*(sinh(7*x)-sinh(5*x))-6*(cosh(4*x)*sinh(x)) +15*cosh(2*x)*sinh(x
  )-10*sinh(x))/32
    .= (1/2*(sinh(7*x)-sinh(5*x))-6*(1/2*(sinh(4*x+1*x)-sinh(4*x-1*x))) +15*
  cosh(2*x)*sinh(x)-10*sinh(x))/32 by Th11
    .= (1/2*(sinh(7*x)-7*sinh(5*x)+sinh(3*x)*6) +15*(cosh(2*x)*sinh(x))-10*
  sinh(x))/32
    .= (1/2*(sinh(7*x)-7*sinh(5*x)+sinh(3*x)*6)+15 *(1/2*(sinh(2*x+1*x)-sinh
  (2*x-1*x)))-10*sinh(x))/32 by Th11
    .= (sinh(7*x)-7*sinh(5*x)+21*sinh(3*x)-35*sinh(x))/64;
  (sinh x)|^8 = (sinh x)|^(7+1)
    .= (sinh(7*x)-7*sinh(5*x)+21*sinh(3*x)-35*sinh(x))/64*sinh(x) by A5,
NEWTON:6
    .= (sinh(7*x)*sinh(x)-7*(sinh(5*x)*sinh x) +21*(sinh(3*x)*sinh(x))-35*(
  sinh(x)*sinh x))/64
    .= (1/2*(cosh(7*x+1*x)-cosh(7*x-1*x))-7*(sinh(5*x)*sinh(x)) +21*(sinh(3*
  x)*sinh(x))-35*(sinh(x)*sinh x))/64 by Th11
    .= (1/2*(cosh(8*x)-cosh(6*x))-(1/2*(cosh(5*x+1*x) -cosh(5*x-1*x))*7)+21*
  (sinh(3*x)*sinh x) -35*(sinh(x)*sinh(x)))/64 by Th11
    .= (1/2*(cosh(8*x)-8*cosh(6*x)+cosh(4*x)*7)+1/2*(cosh(3*x+1*x) -cosh(3*x
  -1*x))*21-35*(sinh(x)*sinh x))/64 by Th11
    .= (1/2*(cosh(8*x)-8*cosh(6*x)+28*cosh(4*x)+-cosh(2*x)*21) -35*(sinh x)
  ^2)/64
    .= (1/2*(cosh(8*x)-8*cosh(6*x)+28*cosh(4*x)+-cosh(2*x)*21) -1/2*(cosh(2*
  x)-1)*35)/64 by Lm7
    .= (cosh(8*x)-8*cosh(6*x)+28*cosh(4*x)-56*cosh(2*x)+35)/128;
  hence thesis by A1,A2,A3,A4,A5;
end;
