reserve a,b,c,k,m,n for Nat;
reserve i,j,x,y for Integer;
reserve p,q for Prime;
reserve r,s for Real;

theorem
  { [x,y] where x,y is positive Nat: 2|^x - 1 = y^2 } = { [1,1] }
  proof
    set A = { [x,y] where x,y is positive Nat: 2|^x - 1 = y^2 };
    thus A c= {[1,1]}
    proof
      let a be object;
      assume a in A;
      then consider x,y being positive Nat such that
A1:   a = [x,y] and
A2:   2|^x - 1 = y^2;
      per cases;
      suppose x > 1;
        then x >= 1+1 by NAT_1:13;
        then consider k being positive Nat such that
A3:     2|^x-1 = 4*k-1 by NUMBER09:11;
A4:     y^2 mod 4 = 0 or y^2 mod 4 = 1 by PYTHTRIP:3,4;
        (4*k-1) mod 4 = ((4*k+0 mod 4) - (1 mod 4)) mod 4 by INT_6:7
        .= (3+(-1)*4) mod 4 by NAT_D:24
        .= 3 mod 4 by NAT_D:61;
        hence thesis by A2,A3,A4,NAT_D:24;
      end;
      suppose x <= 1;
        then
A5:     x = 1 by NAT_1:25;
        then y = 1 by A2,NAT_1:15;
        hence thesis by A1,A5,TARSKI:def 1;
      end;
    end;
    let a be object;
    assume a in {[1,1]};
    then
A6: a = [1,1] by TARSKI:def 1;
    2|^1 - 1 = 1^2;
    hence thesis by A6;
  end;
