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 a on B
proof
A1: A<>B implies ex a st not a on A & not a on B
  proof
    assume A<>B;
    then consider a,b such that
    a on A and
A2: not a on B and
    b on B and
A3: not b on A by Th3;
    consider C such that
A4: a on C & b on C by INCPROJ:def 5;
    consider c,d,e being Element of the Points of IPP such that
A5: c <>d & d<>e & e<>c & c on C & d on C & e on C by INCPROJ:def 7;
    not c on A & not c on B or not d on A & not d on B or not e on A &
    not e on B by A2,A3,A4,A5,INCPROJ:def 4;
    hence thesis;
  end;
  A=B implies ex a st not a on A & not a on B
  proof
    assume
A6: A=B;
    ex a st not a on A by Th1;
    hence thesis by A6;
  end;
  hence thesis by A1;
end;
