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: (x+1)|^3 - x|^3 = y^2 } is infinite
  proof
    set A = { [x,y] where x,y is positive Nat: (x+1)|^3 - x|^3 = y^2 };
    set B = { [x,y] where x,y is positive Nat: 3*x^2 + 3*x - y^2 + 1 = 0 };
    A = B
    proof
      thus A c= B
      proof
        let a be object;
        assume a in A;
        then consider x,y being positive Nat such that
A1:     a = [x,y] & (x+1)|^3 - x|^3 = y^2;
        (x+1)|^3 - x|^3 - y^2 = 3*x^2 + 3*x - y^2 + 1 by Lm13;
        hence thesis by A1;
      end;
      let a be object;
      assume a in B;
      then consider x,y being positive Nat such that
A2:   a = [x,y] & 3*x^2 + 3*x - y^2 + 1 = 0;
      (x+1)|^3 - x|^3 - y^2 = 3*x^2 + 3*x - y^2 + 1 by Lm13;
      hence thesis by A2;
    end;
    hence thesis by Th57;
  end;
