reserve AS for AffinSpace;
reserve A,K,M,X,Y,Z,X9,Y9 for Subset of AS;
reserve zz for Element of AS;
reserve x,y for set;
reserve x,y,z,t,u,w for Element of AS;
reserve K,X,Y,Z,X9,Y9 for Subset of AS;
reserve a,b,c,d,p,q,r,p9 for POINT of IncProjSp_of(AS);
reserve A for LINE of IncProjSp_of(AS);
reserve A,K,M,N,P,Q for LINE of IncProjSp_of(AS);

theorem Th43:
  Y is being_plane & X is being_line & X9 is being_line & X c= Y &
  X9 c= Y & P=[X,1] & Q=[X9,1] implies ex q st q on P & q on Q
proof
  assume that
A1: Y is being_plane and
A2: X is being_line and
A3: X9 is being_line and
A4: X c= Y and
A5: X9 c= Y and
A6: P=[X,1] and
A7: Q=[X9,1];
A8: now
    reconsider q=LDir(X) as POINT of IncProjSp_of(AS) by A2,Th20;
    assume
A9: X // X9;
    take q;
    LDir(X)=LDir(X9) by A2,A3,A9,Th11;
    hence q on P & q on Q by A2,A3,A6,A7,Th30;
  end;
  now
    given y such that
A10: y in X and
A11: y in X9;
    reconsider q=y as Element of the Points of IncProjSp_of(AS) by Th20;
    take q;
    thus q on P & q on Q by A2,A3,A6,A7,A10,A11,Th26;
  end;
  hence thesis by A1,A2,A3,A4,A5,A8,AFF_4:22;
end;
