reserve x for set;
reserve i,j for Integer;
reserve n,n1,n2,n3 for Nat;
reserve p for Prime;
reserve a,b,c,d for Element of GF(p);
reserve K for Ring;
reserve a1,a2,a3,a4,a5,a6 for Element of K;

theorem Th14:
  a1*(a2+a3+a4) = a1*a2+a1*a3+a1*a4
  & a1*(a2+a3-a4) = a1*a2+a1*a3-a1*a4
  & a1*(a2-a3+a4) = a1*a2-a1*a3+a1*a4
  & a1*(a2-a3-a4) = a1*a2-a1*a3-a1*a4
  & a1*(-a2+a3+a4) = -a1*a2+a1*a3+a1*a4
  & a1*(-a2+a3-a4) = -a1*a2+a1*a3-a1*a4
  & a1*(-a2-a3+a4) = -a1*a2-a1*a3+a1*a4
  & a1*(-a2-a3-a4) = -a1*a2-a1*a3-a1*a4
  proof
    thus a1*(a2+a3+a4) = a1*(a2+a3)+a1*a4 by VECTSP_1:def 7
    .= a1*a2+a1*a3+a1*a4 by VECTSP_1:def 7;
    thus a1*(a2+a3-a4) = a1*(a2+a3)+a1*(-a4) by VECTSP_1:def 7
    .= a1*a2+a1*a3+a1*(-a4) by VECTSP_1:def 7
    .= a1*a2+a1*a3-a1*a4 by VECTSP_1:8;
    thus a1*(a2-a3+a4) = a1*(a2+(-a3))+a1*a4 by VECTSP_1:def 7
    .= (a1*a2+a1*(-a3))+a1*a4 by VECTSP_1:def 7
    .= a1*a2-a1*a3+a1*a4 by VECTSP_1:8;
    thus a1*(a2-a3-a4) = a1*(a2+(-a3))+a1*(-a4) by VECTSP_1:def 7
    .= (a1*a2+a1*(-a3))+a1*(-a4) by VECTSP_1:def 7
    .= a1*a2-a1*a3+a1*(-a4) by VECTSP_1:8
    .= a1*a2-a1*a3-a1*a4 by VECTSP_1:8;
    thus a1*(-a2+a3+a4) = a1*(-a2+a3)+a1*a4 by VECTSP_1:def 7
    .= (a1*(-a2)+a1*a3)+a1*a4 by VECTSP_1:def 7
    .= -a1*a2+a1*a3+a1*a4 by VECTSP_1:8;
    thus a1*(-a2+a3-a4) = a1*(-a2+a3)+a1*(-a4) by VECTSP_1:def 7
    .= (a1*(-a2)+a1*a3)+a1*(-a4) by VECTSP_1:def 7
    .= -a1*a2+a1*a3+a1*(-a4) by VECTSP_1:8
    .= -a1*a2+a1*a3-a1*a4 by VECTSP_1:8;
    thus a1*(-a2-a3+a4) = a1*(-a2+(-a3))+a1*a4 by VECTSP_1:def 7
    .= (a1*(-a2)+a1*(-a3))+a1*a4 by VECTSP_1:def 7
    .= -a1*a2+a1*(-a3)+a1*a4 by VECTSP_1:8
    .= -a1*a2-a1*a3+a1*a4 by VECTSP_1:8;
    thus a1*(-a2-a3-a4) = a1*(-a2+(-a3))+a1*(-a4) by VECTSP_1:def 7
    .= (a1*(-a2)+a1*(-a3))+a1*(-a4) by VECTSP_1:def 7
    .= -a1*a2+a1*(-a3)+a1*(-a4) by VECTSP_1:8
    .= -a1*a2-a1*a3+a1*(-a4) by VECTSP_1:8
    .= -a1*a2-a1*a3-a1*a4 by VECTSP_1:8;
  end;
