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
  for m being positive Nat holds
  card { [x,y] where x,y is positive Nat: x^2+y^2+2*x*y-m*x-m*y-m-1 = 0 } = m
  proof
    let m be positive Nat;
    set A = { [x,y] where x,y is positive Nat: x^2+y^2+2*x*y-m*x-m*y-m-1 = 0 };
    set B = { [x,y] where x,y is positive Nat: x+y = m+1 };
    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] and
A2:     x^2+y^2+2*x*y-m*x-m*y-m-1 = 0;
        x^2+y^2+2*x*y-m*x-m*y-m-1 = (x+y-m-1)*(x+y+1);
        then x+y-m-1 = 0 by A2;
        hence thesis by A1;
      end;
      let a be object;
      assume a in B;
      then consider x,y being positive Nat such that
A3:   a = [x,y] & x+y = m+1;
      x^2+y^2+2*x*y-m*x-m*y-m-1 = (x+y-m-1)*(x+y+1);
      hence thesis by A3;
    end;
    hence thesis by Th44;
  end;
