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 Th3:
  A<>B implies ex a,b st a on A & not a on B & b on B & not b on A
proof
  consider a,b,d such that
A1: a<>b and
  b<>d and
  d<>a and
A2: a on A & b on A and
  d on A by INCPROJ:def 7;
  consider r,s,t such that
A3: r<>s and
  s<>t and
  t<>r and
A4: r on B & s on B and
  t on B by INCPROJ:def 7;
  assume
A5: A<>B;
  then
A6: not r on A or not s on A by A3,A4,INCPROJ:def 4;
  not a on B or not b on B by A5,A1,A2,INCPROJ:def 4;
  hence thesis by A2,A4,A6;
end;
