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
  a<>b implies ex A,B st a on A & not a on B & b on B & not b on A
proof
  consider Q such that
A1: a on Q & b on Q by INCPROJ:def 5;
  consider q such that
A2: not q on Q by Th1;
  consider B such that
A3: b on B and
A4: q on B by INCPROJ:def 5;
  assume
A5: a<>b;
  then
A6: not a on B by A1,A2,A3,A4,INCPROJ:def 4;
  consider A such that
A7: a on A and
A8: q on A by INCPROJ:def 5;
  not b on A by A5,A1,A2,A7,A8,INCPROJ:def 4;
  hence thesis by A7,A3,A6;
end;
