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;
reserve x,y,z,t,u,w for Element of AS;
reserve K,X,Y,Z,X9,Y9 for Subset of AS;
reserve a,b,c,d,p,q,r,p9 for POINT of IncProjSp_of(AS);
reserve A for LINE of IncProjSp_of(AS);
reserve A,K,M,N,P,Q for LINE of IncProjSp_of(AS);

theorem
  IncProjSp_of(AS) is 2-dimensional implies AS is AffinPlane
proof
  set x = the Element of AS;
  assume
A1: IncProjSp_of(AS) is 2-dimensional;
  consider X such that
A2: x in X and
  x in X and
  x in X and
A3: X is being_plane by AFF_4:37;
  assume AS is not AffinPlane;
  then consider z such that
A4: not z in X by A3,AFF_4:48;
  set Y=Line(x,z);
A5: Y is being_line by A2,A4,AFF_1:def 3;
  then reconsider A=[PDir(X),2],K=[Y,1] as LINE of IncProjSp_of(AS) by A3,Th23;
  consider a being POINT of IncProjSp_of(AS) such that
A6: a on A and
A7: a on K by A1,INCPROJ:def 9;
  not a is Element of AS by A6,Th27;
  then consider Y9 such that
A8: a=LDir(Y9) and
A9: Y9 is being_line by Th20;
  Y9 '||' Y by A5,A7,A8,A9,Th28;
  then
A10: Y9 // Y by A5,A9,AFF_4:40;
A11: z in Y by AFF_1:15;
A12: x in Y by AFF_1:15;
  Y9 '||' X by A3,A6,A8,A9,Th29;
  then Y c= X by A2,A3,A5,A12,A10,AFF_4:43,56;
  hence contradiction by A4,A11;
end;
