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;
reserve IPP for Fanoian IncProjSp;
reserve a,b,c,d,p,q,r,s for POINT of IPP;
reserve A,B,C,D,L,Q,R,S for LINE of IPP;

theorem Th16:
  ex a,b,c,d,A st a on A & b on A & c on A & d on A & a,b,c,d
  are_mutually_distinct
proof
  consider p,q,r,s,a,b,c,A,B,C,Q,L,R,S,D,d such that
  not q on L and
  not r on L and
  not p on Q and
  not s on Q and
  not p on R and
  not r on R and
  not q on S and
  not s on S and
  {a,p,s} on L and
  {a,q,r} on Q and
  {b,q,s} on R and
  {b,p,r} on S and
A1: {c,p,q} on A and
  {c,r,s} on B and
  {a,b} on C and
  not c on C and
  b on D and
  c on D and
  C,D,R,S are_mutually_distinct and
A2: d on A & c,d,p,q are_mutually_distinct by Lm2;
A3: q on A by A1,INCSP_1:2;
  c on A & p on A by A1,INCSP_1:2;
  hence thesis by A2,A3;
end;
