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;
reserve IPP for Desarguesian 2-dimensional IncProjSp;
reserve c,p,q,x,y for POINT of IPP;
reserve A,B,C,D,K,L,R,X for LINE of IPP;
reserve f for PartFunc of the Points of IPP,the Points of IPP;

theorem Th23:
  for a1,b1,a2,b2 being POINT of IPP holds not p on K & not p on L
  & a1 on K & b1 on K & IncProj(K,p,L).a1=a2 & IncProj(K,p,L).b1=b2 & a2=b2
  implies a1=b1
proof
  let a1,b1,a2,b2 be POINT of IPP;
  assume that
A1: not p on K and
A2: not p on L and
A3: a1 on K and
A4: b1 on K and
A5: IncProj(K,p,L).a1=a2 and
A6: IncProj (K,p,L).b1=b2 and
A7: a2=b2;
  reconsider a2=IncProj(K,p,L).a1 as POINT of IPP by A5;
A8: a2 on L by A1,A2,A3,Th20;
  then consider A such that
A9: p on A and
A10: a1 on A and
A11: a2 on A by A1,A2,A3,Def1;
  reconsider b2=IncProj(K,p,L).b1 as Element of the Points of IPP by A6;
  b2 on L by A1,A2,A4,Th20;
  then consider B such that
A12: p on B and
A13: b1 on B and
A14: b2 on B by A1,A2,A4,Def1;
  A=B by A2,A5,A6,A7,A8,A9,A11,A12,A14,INCPROJ:def 4;
  hence thesis by A1,A3,A4,A9,A10,A13,INCPROJ:def 4;
end;
