reserve x for set;
reserve i,j for Integer;
reserve n,n1,n2,n3 for Nat;
reserve K,K1,K2,K3 for Field;
reserve SK1,SK2 for Subfield of K;
reserve ek,ek1,ek2 for Element of K;
reserve p for Prime;
reserve a,b,c for Element of GF(p);
reserve F for FinSequence of GF(p);
reserve Px,Py,Pz for Element of GF(p);

theorem Th41:
  for X,Y,Z be Element of GF(p) holds
  EC_WEqProjCo(a,b,p).([X,Y,Z]) = Y |^2*Z-(X|^3 +a*X*Z |^2+b*Z |^3)
  proof
    let X,Y,Z be Element of GF(p);
    set DX = [:the carrier of GF(p),
    the carrier of GF(p), the carrier of GF(p):];
    reconsider P = [X,Y,Z] as Element of DX;
    P`1_3 = X & P`2_3 = Y & P`3_3 = Z;
    hence thesis by Def8;
  end;
