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 Th21:
  for gi, gj, gij, a be Element of GF(p) st
  gi = i mod p & gj = j mod p & gij = (i-j) mod p holds
  gi*a - gj*a = gij*a
  proof
    let gi, gj, gij, a be Element of GF(p) such that
    A1: gi = i mod p & gj = j mod p & gij = (i-j) mod p;
    gj + gij = (j+(i-j)) mod p by A1,EC_PF_1:15
    .= gi by A1;
    then gj*a + gij*a - gj*a = gi*a - gj*a by VECTSP_1:def 7;
    then gij*a + (gj*a + (-gj*a)) = gi*a - gj*a by ALGSTR_1:7;
    then gij*a + 0.GF(p) = gi*a - gj*a by VECTSP_1:19;
    hence thesis by ALGSTR_1:7;
  end;
