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 Th15:
  G is IncProjSp implies ex a,b,c,d st a,b,c,d is_a_quadrangle
proof
  assume
A1: G is IncProjSp;
  then consider A,B such that
A2: A<>B by Th7;
  consider a,b such that
A3: a on A & a|'B and
A4: b on B & b|'A by A1,A2,PROJRED1:3;
  consider q such that
A5: q on B & q|'A and
A6: b<>q by A1,A3,Th9;
A7: {b,q} on B & b,q|'A by A4,A5,INCSP_1:1;
A8: G is configuration by A1,Th4;
  consider p such that
A9: p on A & p|'B and
A10: a<>p by A1,A3,Th9;
  take a,p,b,q;
  {a,p} on A & a,p|'B by A3,A9,INCSP_1:1;
  hence thesis by A10,A6,A8,A7,Th14;
end;
