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

theorem
  (cosh x)|^3 = (cosh(3*x)+3*cosh x)/4 & (cosh x)|^4 = (cosh(4*x)+4*cosh
(2*x)+3)/8 & (cosh x)|^5 = (cosh(5*x)+5*cosh(3*x)+10*cosh x)/16 & (cosh x)|^6 =
(cosh(6*x)+6*cosh(4*x)+15*cosh(2*x)+10)/32 & (cosh x)|^7 = (cosh(7*x)+7*cosh(5*
x)+21*cosh(3*x)+35*cosh x)/64 & (cosh x)|^8 = (cosh(8*x)+8*cosh(6*x)+28*cosh(4*
  x)+56*cosh(2*x)+35)/128
proof
A1: (cosh x)|^3 = (cosh x)|^(1+1+1) .= (cosh x)|^(1+1)*cosh(x) by NEWTON:6
    .= (cosh x)|^1*cosh(x)*cosh(x) by NEWTON:6
    .= ((cosh x)^2*cosh(x) + 3*(cosh(x)*((cosh x)^2-(sinh x)^2 +(sinh x)^2))
  )/4
    .= ((cosh x)^2*cosh(x)+3*(cosh(x)*(1+(sinh x)^2)))/4 by Lm3
    .= (cosh(x)*(cosh(x)*cosh(x)+(sinh x)^2)+2*(cosh(x)*(sinh x)^2) + 3*cosh
  x)/4
    .= (cosh(x)*cosh(x+x)+2*(cosh(x)*(sinh x)^2) + 3*cosh x)/4 by Lm10
    .= (cosh(x)*cosh(2*x)+2*sinh(x)*cosh(x)*sinh(x)+3*cosh(x))/4
    .= (cosh(2*x)*cosh(x)+sinh(2*x)*sinh(x)+3*cosh(x))/4 by Th26
    .= (cosh(x+x+x)+3*cosh x)/4 by Lm10
    .= (cosh(3*x)+3*cosh x)/4;
  then
A2: (cosh x)|^4 = (cosh(3*x)+3*cosh x)/4*cosh(x) by POLYEQ_2:4
    .= (cosh(3*x)*cosh(x)+3*(cosh(x)*cosh x))/4
    .= (1/2*(cosh(3*x+1*x)+cosh(3*x-x))+3*(cosh(x)*cosh x))/4 by Th11
    .= (1/2*(cosh(4*x)+cosh(2*x))+3*(1/2*(cosh(x+x)+cosh(x-x))))/4 by Th11
    .= (1/2*(cosh(4*x)+cosh(2*x))+1/2*(cosh(2*x)+1)*3)/4 by SIN_COS2:15,def 4
    .= (cosh(4*x)+4*cosh(2*x)+3)/8;
A3: (cosh x)|^5 = (cosh x)|^(4+1)
    .= (cosh(4*x)+4*cosh(2*x)+3)/8*cosh(x) by A2,NEWTON:6
    .= (cosh(4*x)*cosh(x)+4*(cosh(2*x)*cosh(x))+3*cosh(x))/8
    .= (1/2*(cosh(4*x+1*x)+cosh(4*x-x)) + 4*(cosh(2*x)*cosh(x)) + 3*cosh(x))
  /8 by Th11
    .= (1/2*(cosh(5*x)+cosh(3*x)) + 4*(1/2*(cosh(2*x+1*x)+cosh(2*x-x))) + 3*
  cosh(x))/8 by Th11
    .= (cosh(5*x)+5*cosh(3*x)+10*cosh(x))/16;
A4: (cosh x)|^6 = (cosh x)|^(5+1)
    .= (cosh(5*x)+5*cosh(3*x)+10*cosh(x))/16*cosh(x) by A3,NEWTON:6
    .= (cosh(5*x)*cosh(x)+5*(cosh(3*x)*cosh(x))+10*(cosh(x)*cosh(x)))/16
    .= (1/2*(cosh(5*x+x)+cosh(5*x-x))+5*(cosh(3*x)*cosh(x)) + 10*(cosh(x)*
  cosh x))/16 by Th11
    .= (1/2*(cosh(5*x+x)+cosh(5*x-x)) + 1/2*(cosh(3*x+x)+cosh(3*x-x))*5 + 10
  *(cosh(x)*cosh x))/16 by Th11
    .= (1/2*(cosh(6*x)+6*cosh(4*x)+cosh(2*x)*5) + 10*(cosh 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: (cosh x)|^7 = (cosh(x))|^(6+1)
    .= (cosh(6*x)+6*cosh(4*x)+15*cosh(2*x)+10)/32*cosh(x) by A4,NEWTON:6
    .= (cosh(6*x)*cosh(x) + 6*(cosh(4*x)*cosh(x)) + 15*(cosh(2*x)*cosh(x)) +
  10*cosh(x))/32
    .= (1/2*(cosh(6*x+1*x)+cosh(6*x-1*x)) + 6*(cosh(4*x)*cosh(x)) + 15*(cosh
  (2*x)*cosh(x)) + 10*cosh(x))/32 by Th11
    .= (1/2*(cosh(7*x)+cosh(5*x)) + 1/2*(cosh(4*x+x)+cosh(4*x-x))*6 + 15*(
  cosh(2*x)*cosh(x)) + 10*cosh(x))/32 by Th11
    .= (1/2*(cosh(7*x)+7*cosh(5*x)+cosh(3*x)*6) + 1/2*(cosh(2*x+x)+cosh(2*x-
  x))*15 + 10*cosh(x))/32 by Th11
    .= (cosh(7*x)+7*cosh(5*x)+21*cosh(3*x)+35*cosh(x))/64;
  (cosh x)|^8 = (cosh x)|^(7+1)
    .= (cosh(7*x)+7*cosh(5*x)+21*cosh(3*x)+35*cosh(x))/64*cosh(x) by A5,
NEWTON:6
    .= (cosh(7*x)*cosh(x) + 7*cosh(5*x)*cosh(x) + 21*cosh(3*x)*cosh(x) + 35*
  (cosh x)^2)/64
    .= (1/2*(cosh(7*x+1*x)+cosh(7*x-x)) + 7*cosh(5*x)*cosh(x) + 21*cosh(3*x)
  *cosh(x) + 35*(cosh x)^2)/64 by Th11
    .= (1/2*(cosh(8*x)+cosh(6*x)) + 7*(cosh(5*x)*cosh(x)) + 21*cosh(3*x)*
  cosh(x) + 35*(cosh x)^2)/64
    .= (1/2*(cosh(8*x)+cosh(6*x)) + 7*(1/2*(cosh(5*x+1*x)+cosh(5*x-1*x))) +
  21*cosh(3*x)*cosh(x) + 35*(cosh x)^2)/64 by Th11
    .= (1/2*(cosh(8*x)+8*cosh(6*x)+cosh(4*x)*7) + 21*(cosh(3*x)*cosh x) + 35
  *(cosh x)^2)/64
    .= (1/2*(cosh(8*x)+8*cosh(6*x)+cosh(4*x)*7) + 21*(1/2*(cosh(3*x+1*x)+
  cosh(3*x-x))) + 35*(cosh x)^2)/64 by Th11
    .= (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;
