reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;
reserve u,v for non zero Element of TOP-REAL 3;
reserve P,Q,R for POINT of IncProjSp_of real_projective_plane,
            L for LINE of IncProjSp_of real_projective_plane,
        p,q,r for Point of real_projective_plane;

theorem Th62:
  IncProjSp_of real_projective_plane is IncProjectivePlane
  proof
    IncProjSp_of real_projective_plane is 2-dimensional
    proof
      for M,N being LINE of IncProjSp_of real_projective_plane
      ex q being POINT of IncProjSp_of real_projective_plane st
      q on M & q on N
      proof
        let M,N be LINE of IncProjSp_of real_projective_plane;
        consider p1,q1 be Point of real_projective_plane such that
        p1 <> q1 and
A1:     M = Line(p1,q1) by Th60;
        consider p2,q2 be Point of real_projective_plane such that
        p2 <> q2 and
A2:     N = Line(p2,q2) by Th60;
        consider q be Element of real_projective_plane such that
A3:     p1,q1,q are_collinear and
A4:     p2,q2,q are_collinear by ANPROJ_2:def 14;
        reconsider Q = q as POINT of IncProjSp_of real_projective_plane
          by INCPROJ:3;
        take Q;
        thus thesis by A1,A2,A3,A4,Th61;
      end;
      hence thesis by INCPROJ:def 9;
    end;
    hence thesis;
  end;
