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 Th26:
  for g2, a, b be Element of GF(p) st g2 = 2 mod p
  holds (a - b) |^2 = a |^2 - g2*a*b + b |^2
  proof
    let g2, a, b be Element of GF(p) such that
    A1: g2 = 2 mod p;
    thus (a - b) |^2 = (a - b)*(a - b) by EC_PF_1:22
    .= a*(a-b)-b*(a-b) by VECTSP_1:13
    .= a*a-a*b-b*(a-b) by VECTSP_1:11
    .= a |^2 - a*b - b*(a-b) by EC_PF_1:22
    .= a |^2 - a*b - (b*a-b*b) by VECTSP_1:11
    .= a |^2 - a*b - (a*b-b |^2) by EC_PF_1:22
    .= a |^2 + (-a*b-(a*b-b |^2)) by ALGSTR_1:7
    .= a |^2 + (-a*b+(-a*b+b |^2)) by VECTSP_1:17
    .= a |^2 + ((-a*b-a*b)+b |^2) by ALGSTR_1:7
    .= a |^2 + (g2*(-a*b)+b |^2) by A1,Th20
    .= a |^2 + g2*(-a*b) + b|^2 by ALGSTR_1:7
    .= a |^2 - g2*(a*b) + b |^2 by VECTSP_1:8
    .= a |^2 - g2*a*b + b |^2 by GROUP_1:def 3;
  end;
