reserve IPP for IncProjSp;
reserve a,b,c,d,p,q,o,r,s,t,u,v,w,x,y for POINT of IPP;
reserve A,B,C,D,L,P,Q,S for LINE of IPP;

theorem
  ex A,B,C st a on A & a on B & a on C & A<>B & B<>C & C<>A
proof
  consider Q such that
A1: not a on Q by Th2;
  consider b1,b2,b3 being Element of the Points of IPP such that
A2: b1<>b2 and
A3: b2<>b3 and
A4: b3<>b1 and
A5: b1 on Q and
A6: b2 on Q and
A7: b3 on Q by INCPROJ:def 7;
  consider B1 being LINE of IPP such that
A8: a on B1 & b1 on B1 by INCPROJ:def 5;
A9: not b3 on B1 by A1,A4,A5,A7,A8,INCPROJ:def 4;
  consider B2 being LINE of IPP such that
A10: a on B2 & b2 on B2 by INCPROJ:def 5;
  consider B3 being Element of the Lines of IPP such that
A11: a on B3 & b3 on B3 by INCPROJ:def 5;
A12: not b2 on B3 by A1,A3,A6,A7,A11,INCPROJ:def 4;
  not b1 on B2 by A1,A2,A5,A6,A10,INCPROJ:def 4;
  hence thesis by A8,A10,A11,A12,A9;
end;
