reserve p for 5_or_greater Prime;
reserve z for Element of EC_WParam p;

theorem ThAffCo:
  for p be 5_or_greater Prime,
      z be Element of EC_WParam p,
      P be Element of EC_SetProjCo(z`1,z`2,p)
  holds rep_pt(P) is Element of EC_SetAffCo(z,p)
  proof
    let p be 5_or_greater Prime,
        z be Element of EC_WParam p,
        P be Element of EC_SetProjCo(z`1,z`2,p);
    set a= z`1;
    set b= z`2;
    consider PP be Element of ProjCo(GF(p)) such that
    A2: PP = P & PP in EC_SetProjCo(a,b,p);
    consider Q be Element of ProjCo(GF(p)) such that
    A3: Q = rep_pt(P);
    reconsider Q as Element of EC_SetProjCo(a,b,p) by A3,EC_PF_2:36;
    per cases;
    suppose PP`3_3 = 0;
      then B1: Q = [0,1,0] by A2,A3,EC_PF_2:def 7;
      Q in {QQ where QQ is Element of EC_SetProjCo(a,b,p) :
      QQ`3_3 = 1 or QQ = [0,1,0] } by B1;
      hence thesis by A3;
    end;
    suppose PP`3_3 <> 0;
      then rep_pt(PP) = [(PP`1_3)*(PP`3_3)",(PP`2_3)*(PP`3_3)",1]
      by EC_PF_2:def 7;
      then B1: Q`3_3 = 1 by A2,A3,EC_PF_2:def 5;
      Q in {QQ where QQ is Element of EC_SetProjCo(a,b,p) :
      QQ`3_3 = 1 or QQ = [0,1,0] } by B1;
      hence thesis by A3;
    end;
  end;
