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
  ex a,A,B,C,D st a on A & a on B & a on C & a on D & 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
A1: {b,q,s} on R & {b,p,r} on S and
  {c,p,q} on A and
  {c,r,s} on B and
A2: {a,b} on C and
  not c on C and
A3: b on D and
  c on D and
A4: C,D,R,S are_mutually_distinct and
  d on A and
  c,d,p,q are_mutually_distinct by Lm2;
A5: b on C by A2,INCSP_1:1;
  b on S & b on R by A1,INCSP_1:2;
  hence thesis by A3,A4,A5;
end;
