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 Th22:
  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) implies for P ex a,b,c
  st a,b,c are_mutually_distinct & {a,b,c} on P
proof
  assume that
A1: G is configuration and
A2: for p,q ex M st {p,q} on M and
A3: for P,Q ex a st a on P,Q and
A4: ex a,b,c,d st a,b,c,d is_a_quadrangle;
  hereby
    let P;
    consider a,b,c,d such that
A5: a,b,c,d is_a_quadrangle by A4;
A6: a,b,c is_a_triangle by A5;
    thus ex a,b,c st a,b,c are_mutually_distinct & {a,b,c} on P
    proof
      now
        per cases by A6,Th5;
        case
A7:       a|'P;
          consider B such that
A8:       {a,b} on B by A2;
          consider D such that
A9:       {a,d} on D by A2;
          consider C such that
A10:      {a,c} on C by A2;
A11:      a on D by A9,INCSP_1:1;
A12:      a on C by A10,INCSP_1:1;
          a on B by A8,INCSP_1:1;
          then
A13:      a on B,C,D by A12,A11;
          consider p such that
A14:      p on P,B by A3;
A15:      B,C,D are_mutually_distinct by A5,A8,A10,A9,Th20;
          consider q such that
A16:      q on P,C by A3;
A17:      q on P by A16;
          consider r such that
A18:      r on P,D by A3;
A19:      r on P by A18;
          p on P by A14;
          then {p,q,r} on P by A17,A19,INCSP_1:2;
          hence thesis by A1,A7,A13,A15,A14,A16,A18,Th21;
        end;
        case
A20:      b|'P;
          consider B such that
A21:      {b,a} on B by A2;
          consider D such that
A22:      {b,d} on D by A2;
          consider C such that
A23:      {b,c} on C by A2;
A24:      b on D by A22,INCSP_1:1;
A25:      b on C by A23,INCSP_1:1;
          b on B by A21,INCSP_1:1;
          then
A26:      b on B,C,D by A25,A24;
          consider q such that
A27:      q on P,C by A3;
          b,a,c,d is_a_quadrangle by A5,Th13;
          then
A28:      B,C,D are_mutually_distinct by A21,A23,A22,Th20;
          consider p such that
A29:      p on P,B by A3;
A30:      q on P by A27;
          consider r such that
A31:      r on P,D by A3;
A32:      r on P by A31;
          p on P by A29;
          then {p,q,r} on P by A30,A32,INCSP_1:2;
          hence thesis by A1,A20,A26,A28,A29,A27,A31,Th21;
        end;
        case
A33:      c|'P;
          consider B such that
A34:      {c,a} on B by A2;
          consider D such that
A35:      {c,d} on D by A2;
          consider C such that
A36:      {c,b} on C by A2;
A37:      c on D by A35,INCSP_1:1;
A38:      c on C by A36,INCSP_1:1;
          c on B by A34,INCSP_1:1;
          then
A39:      c on B,C,D by A38,A37;
          consider q such that
A40:      q on P,C by A3;
          c,a,b,d is_a_quadrangle by A5,Th13;
          then
A41:      B,C,D are_mutually_distinct by A34,A36,A35,Th20;
          consider p such that
A42:      p on P,B by A3;
A43:      q on P by A40;
          consider r such that
A44:      r on P,D by A3;
A45:      r on P by A44;
          p on P by A42;
          then {p,q,r} on P by A43,A45,INCSP_1:2;
          hence thesis by A1,A33,A39,A41,A42,A40,A44,Th21;
        end;
      end;
      hence thesis;
    end;
  end;
end;
