reserve IPS for IncProjSp,
  z for POINT of IPS;
reserve IPP for Desarguesian 2-dimensional IncProjSp,
  a,b,c,d,p,pp9,q,o,o9,o99 ,oo9 for POINT of IPP,
  r,s,x,y,o1,o2 for POINT of IPP,
  O1,O2,O3,O4,A,B,C,O,Q,Q1 ,Q2,Q3,R,S,X for LINE of IPP;

theorem Th3:
  not o on A & not o on B & y on B implies ex x st x on A & IncProj
  (A,o,B).x = y
proof
  consider X such that
A1: o on X & y on X by INCPROJ:def 5;
  consider x such that
A2: x on X and
A3: x on A by INCPROJ:def 9;
  assume ( not o on A)& not o on B & y on B;
  then IncProj(A,o,B).x = y by A1,A2,A3,PROJRED1:def 1;
  hence thesis by A3;
end;
