reserve G for IncProjStr;
reserve a,a1,a2,b,b1,b2,c,d,p,q,r for POINT of G;
reserve A,B,C,D,M,N,P,Q,R for LINE of G;

theorem Th23:
  G is IncProjectivePlane iff G is configuration & (for p,q ex M
  st {p,q} on M) & (for P,Q ex a st a on P,Q) & ex a,b,c,d st a,b,c,d
  is_a_quadrangle
proof
  hereby
    assume
A1: G is IncProjectivePlane;
    hence G is configuration by Th4;
    thus for p,q ex M st {p,q} on M by A1,Th4;
    thus for P,Q ex a st a on P,Q
    proof
      let P,Q;
      consider a such that
A2:   a on P & a on Q by A1,INCPROJ:def 9;
      take a;
      thus thesis by A2;
    end;
    thus ex a,b,c,d st a,b,c,d is_a_quadrangle by A1,Th15;
  end;
  hereby
    assume that
A3: G is configuration and
A4: for p,q ex M st {p,q} on M and
A5: for P,Q ex a st a on P,Q and
A6: ex a,b,c,d st a,b,c,d is_a_quadrangle;
    ex a,b,c st a,b,c is_a_triangle by A6;
    then
A7: ex p,P st p|'P by A4,Th17;
    ( for P ex a,b,c st a,b,c are_mutually_distinct & {a,b,c} on P)& for
a,b,c,d, p,M,N,P,Q st {a,b,p} on M & {c,d,p} on N & {a,c} on P & {b,d} on Q & p
    |'P & p|' Q & M<>N holds ex q st q on P,Q by A3,A4,A5,A6,Th22;
    then reconsider G9=G as IncProjSp by A3,A4,A7,Th4;
    for P,Q being LINE of G9 ex a being POINT of G9 st a on P & a on Q
    proof
      let P,Q be LINE of G9;
      consider a being POINT of G9 such that
A8:   a on P,Q by A5;
      take a;
      thus thesis by A8;
    end;
    hence G is IncProjectivePlane by INCPROJ:def 9;
  end;
end;
