reserve p,q,r,th,th1 for Real;
reserve n for Nat;

theorem
  (cosh.p + sinh.p) |^ n = cosh.(n*p) + sinh.(n*p)
proof
  defpred X[Nat] means for p holds (cosh.p + sinh.p) |^ $1 = cosh.(
  $1*p) + sinh.($1*p);
A1: for n st X[n] holds X[n+1]
  proof
    let n such that
A2: for p holds (cosh.p + sinh.p) |^ n = cosh.(n*p) + sinh.(n*p);
    for p holds (cosh.p + sinh.p) |^ (n+1) = cosh.((n+1)*p) + sinh.((n+1)* p)
    proof
      let p;
A3:   (cosh.(n*p))*(cosh.p)+(sinh.(n*p))*(sinh.p) =((exp_R.(n*p) + exp_R.(
      -n*p))/2)*(cosh.p) +(sinh.(n*p))*(sinh.p) by Def3
        .=((exp_R.(n*p) + exp_R.(-n*p))/2)*((exp_R.p + exp_R.-p)/2) +(sinh.(
      n*p))*(sinh.p) by Def3
        .=((exp_R.(n*p) + exp_R.(-n*p))/2)*((exp_R.p + exp_R.-p)/2) +((exp_R
      .(n*p) - exp_R.(-n*p))/2)*(sinh.p) by Def1
        .=((exp_R.(n*p))/2 + (exp_R.(-n*p))/2)*((exp_R.p)/2 + (exp_R.-p)/2)
      +((exp_R.(n*p))/2 - (exp_R.(-n*p))/2)*((exp_R.p - exp_R.-p)/2) by Def1
        .=2*( ( (exp_R.(n*p))*(exp_R.p) )/(2*2) ) +2*(((exp_R.(-n*p))/2)*((
      exp_R.-p)/2))
        .=2*( exp_R.(p*n+p*1)/(2*2) ) +2*( ( (exp_R.(-n*p))*(exp_R.-p) )/(2*
      2) ) by Th12
        .=2*( exp_R.(p*(n+1))/(2*2) ) +2*( exp_R.(-n*p+-p)/(2*2) ) by Th12
        .=(exp_R.(p*(n+1))+exp_R.(-(p*(n+1))))/2
        .=cosh.(p*(n+1)) by Def3;
A4:   (cosh.(n*p))*(sinh.p)+(sinh.(n*p))*(cosh.p) =((exp_R.(n*p) + exp_R.(
      -n*p))/2)*(sinh.p) +(sinh.(n*p))*(cosh.p) by Def3
        .=((exp_R.(n*p) + exp_R.(-n*p))/2)*((exp_R.p - exp_R.-p)/2) +(sinh.(
      n*p))*(cosh.p) by Def1
        .=((exp_R.(n*p) + exp_R.(-n*p))/2)*((exp_R.p - exp_R.-p)/2) +((exp_R
      .(n*p) - exp_R.(-n*p))/2)*(cosh.p) by Def1
        .=((exp_R.(n*p))/2)*((exp_R.p)/2) - ((exp_R.(n*p))/2)*((exp_R.-p)/2)
+((exp_R.(-n*p))/2)*((exp_R.p)/2) - ((exp_R.(-n*p))/2)*((exp_R.-p)/2) +((exp_R.
      (n*p) - exp_R.(-n*p))/2)*((exp_R.p + exp_R.-p)/2) by Def3
        .=2*(( (exp_R.(n*p))*(exp_R.p) )/4 ) +2*( -( ((exp_R.(-n*p))*(exp_R.
      -p))/(2*2) ))
        .=2*( exp_R.(n*p+p)/4 ) +2*( -( ((exp_R.(-n*p))*(exp_R.-p))/4 )) by
Th12
        .=2*( exp_R.(n*p+p)/4 ) +2*( -( (exp_R.(-n*p+-p))/4 ) ) by Th12
        .=( exp_R.(p*(n+1))- exp_R.(-(p*(n+1))) )/2
        .=sinh.(p*(n+1)) by Def1;
      (cosh.p + sinh.p) |^ (n+1) =(cosh.p + sinh.p) |^ n * (cosh.p + sinh.
      p) by NEWTON:6
        .=(cosh.(n*p) + sinh.(n*p)) * (cosh.p + sinh.p) by A2
        .=sinh.((n+1)*p)+cosh.((n+1)*p) by A3,A4;
      hence thesis;
    end;
    hence thesis;
  end;
A5: X[0] by Th15,Th16,NEWTON:4;
  for n holds X[n] from NAT_1:sch 2(A5,A1);
  hence thesis;
end;
