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

theorem
  for p,q,r being Prime st p*(p+1) + q*(q+1) = r*(r+1) holds p = q = 2 & r = 3
  proof
    let p,q,r be Prime;
    assume p*(p+1) + q*(q+1) = r*(r+1);
    then per cases by Th44;
    suppose p = 2 & q = 2 & r = 3;
      hence thesis;
    end;
    suppose p = 5 & q = 3 & r = 6;
      hence thesis by XPRIMES0:6;
    end;
    suppose p = 3 & q = 5 & r = 6;
      hence thesis by XPRIMES0:6;
    end;
  end;
