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
  ProjHorizon(AS) is IncProjSp implies AS is not AffinPlane
proof
  set XX=ProjHorizon(AS);
  assume ProjHorizon(AS) is IncProjSp;
  then consider
  a being Element of the Points of XX, A being Element of the Lines
  of XX such that
A1: not a on A by INCPROJ:def 6;
  consider X such that
A2: a=LDir(X) and
A3: X is being_line by Th14;
  consider Y such that
A4: A=PDir(Y) and
A5: Y is being_plane by Th15;
  assume AS is AffinPlane;
  then Y = the carrier of AS by A5,Th2;
  then X '||' Y by A3,A5,AFF_4:42;
  hence contradiction by A1,A2,A3,A4,A5,Th36;
end;
