reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;
reserve z for Complex;

theorem
  n >= 2 implies ex k being positive Nat st 2|^n-1 = 4*k-1
  proof
    defpred P[Nat] means
    $1 >= 2 implies ex k being positive Nat st 2|^$1-1 = 4*k-1;
A1: P[2]
    proof
      assume 2 >= 2;
      take 1;
      thus thesis by Lm2;
    end;
A2: for j being Nat st 2 <= j holds P[j] implies P[j+1]
    proof
      let j be Nat such that
A3:   2 <= j;
      assume P[j];
      then consider k being positive Nat such that
A4:   2|^j-1 = 4*k-1 by A3;
      assume j+1 >= 2;
      take 2*k;
      2|^(j+1) = 2|^j*2 by NEWTON:6;
      hence thesis by A4;
    end;
    for i being Nat st 2 <= i holds P[i] from NAT_1:sch 8(A1,A2);
    hence thesis;
  end;
