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 Th2:
  ex A st not a on A
proof
  consider p,L such that
A1: not p on L by INCPROJ:def 6;
  now
    now
      consider q,r,s such that
      q<>r and
      r<>s and
A2:   s<>q and
A3:   q on L and
      r on L and
A4:   s on L by INCPROJ:def 7;
      assume
A5:   a on L;
A6:   now
        consider S such that
A7:     s on S & p on S by INCPROJ:def 5;
        assume a<>s;
        then not a on S by A1,A5,A4,A7,INCPROJ:def 4;
        hence thesis;
      end;
      now
        consider S such that
A8:     q on S & p on S by INCPROJ:def 5;
        assume a<>q;
        then not a on S by A1,A5,A3,A8,INCPROJ:def 4;
        hence thesis;
      end;
      hence thesis by A2,A6;
    end;
    hence thesis;
  end;
  hence thesis;
end;
