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 st not a on A & not b on A
proof
  now
    consider A such that
A1: a on A & b on A by INCPROJ:def 5;
    consider c such that
A2: c on A & c <>a & c <>b by Th8;
    consider d such that
A3: not d on A by Th1;
    consider B such that
A4: d on B & c on B by INCPROJ:def 5;
    ( not a on B)& not b on B by A1,A2,A3,A4,INCPROJ:def 4;
    hence thesis;
  end;
  hence thesis;
end;
