reserve AS for AffinSpace;
reserve A,K,M,X,Y,Z,X9,Y9 for Subset of AS;
reserve zz for Element of AS;
reserve x,y for set;

theorem Th1:
  AS is AffinPlane & X=the carrier of AS implies X is being_plane
proof
  assume that
A1: AS is AffinPlane and
A2: X=the carrier of AS;
  consider a,b,c being Element of AS such that
A3: not LIN a,b,c by AFF_1:12;
  set P=Line(a,b),K=Line(a,c);
A4: b in P by AFF_1:15;
A5: c in K by AFF_1:15;
  a<>b by A3,AFF_1:7;
  then
A6: P is being_line by AFF_1:def 3;
  set Y=Plane(K,P);
A7: a in P by AFF_1:15;
  a<>c by A3,AFF_1:7;
  then
A8: K is being_line by AFF_1:def 3;
A9: a in K by AFF_1:15;
A10: not K // P
  proof
    assume K // P;
    then c in P by A7,A9,A5,AFF_1:45;
    hence contradiction by A3,A7,A4,A6,AFF_1:21;
  end;
  now
    let x be object;
    assume x in X;
    then reconsider a=x as Element of AS;
    set K9=a*K;
A11: K9 is being_line by A8,AFF_4:27;
A12: K // K9 by A8,AFF_4:def 3;
    then not K9 // P by A10,AFF_1:44;
    then consider b being Element of AS such that
A13: b in K9 and
A14: b in P by A1,A6,A11,AFF_1:58;
    a in K9 by A8,AFF_4:def 3;
    then a,b // K by A12,A13,AFF_1:40;
    then a in {zz: ex b being Element of AS st zz,b // K & b in P} by A14;
    hence x in Y by AFF_4:def 1;
  end;
  then
A15: X c= Y;
  Y is being_plane by A6,A8,A10,AFF_4:def 2;
  hence thesis by A2,A15,XBOOLE_0:def 10;
end;
