reserve n for Nat,
  a,b,c,d for Real,
  s for Real_Sequence;

theorem
  a<>0 implies ((1/a)|^(n+1)+a|^(n+1))|^2 = (1/a)|^(2*n+2)+a|^(2*n+2)+2
proof
  assume
A1: a<>0;
  ((1/a)|^(n+1)+a|^(n+1))|^(1+1) =((1/a)|^(n+1)+a|^(n+1))|^1*((1/a)|^(n+1)
  +a|^(n+1)) by NEWTON:6
    .=((1/a)|^(n+1)+a|^(n+1))*((1/a)|^(n+1)+a|^(n+1))
    .=((1/a)|^(n+1)*(1/a)|^(n+1)+(1/a)|^(n+1)*a|^(n+1)) +(a|^(n+1)*(1/a)|^(n
  +1)+a|^(n+1)*a|^(n+1))
    .=((1/a*(1/a))|^(n+1)+(1/a)|^(n+1)*a|^(n+1)) +(a|^(n+1)*(1/a)|^(n+1)+a|^
  (n+1)*a|^(n+1)) by NEWTON:7
    .=((1/a*(1/a))|^(n+1)+(1/a*a)|^(n+1)) +(a|^(n+1)*(1/a)|^(n+1)+a|^(n+1)*a
  |^(n+1)) by NEWTON:7
    .=((1/a*(1/a))|^(n+1)+(1/a*a)|^(n+1)) +((a*(1/a))|^(n+1)+a|^(n+1)*a|^(n+
  1)) by NEWTON:7
    .=((1/a*(1/a))|^(n+1)+(1/a*a)|^(n+1)) +((a*(1/a))|^(n+1)+(a*a)|^(n+1))
  by NEWTON:7
    .=((1/a*(1/a))|^(n+1)+1|^(n+1))+((1/a*a)|^(n+1)+(a*a)|^(n+1)) by A1,
XCMPLX_1:106
    .=((1/a*(1/a))|^(n+1)+1|^(n+1))+(1|^(n+1)+(a*a)|^(n+1)) by A1,XCMPLX_1:106
    .=((1/a*(1/a))|^(n+1)+1)+(1|^(n+1)+(a*a)|^(n+1))
    .=((1/a*(1/a))|^(n+1)+1)+(1+(a*a)|^(n+1))
    .=(((1/a)*(1/a))|^(n+1)+(a*a)|^(n+1))+2
    .=((1/a)*(1/a))|^(n+1)+(a|^2)|^(n+1)+2 by WSIERP_1:1
    .=((1/a)*(1/a))|^(n+1)+a|^(2*(n+1))+2 by NEWTON:9
    .=((1/a)|^2)|^(n+1)+a|^(2*n+2)+2 by WSIERP_1:1
    .=(1/a)|^(2*(n+1))+a|^(2*n+2)+2 by NEWTON:9
    .=(1/a)|^(2*n+2)+a|^(2*n+2)+2;
  hence thesis;
end;
