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);

theorem Th36:
  for a being Element of the Points of ProjHorizon(AS),A being
  Element of the Lines of ProjHorizon(AS) st a=LDir(X) & A=PDir(Y) & X is
  being_line & Y is being_plane holds (a on A iff X '||' Y)
proof
  let a be Element of the Points of ProjHorizon(AS),A be Element of the Lines
  of ProjHorizon(AS) such that
A1: a=LDir(X) and
A2: A=PDir(Y) and
A3: X is being_line and
A4: Y is being_plane;
A5: now
    assume a on A;
    then [a,A] in the Inc of ProjHorizon(AS) by INCSP_1:def 1;
    then consider X9,Y9 such that
A6: a=LDir(X9) and
A7: A=PDir(Y9) and
A8: X9 is being_line and
A9: Y9 is being_plane and
A10: X9 '||' Y9 by Def12;
    X // X9 by A1,A3,A6,A8,Th11;
    then
A11: X '||' Y9 by A10,AFF_4:56;
    Y9 '||' Y by A2,A4,A7,A9,Th13;
    hence X '||' Y by A9,A11,AFF_4:59,60;
  end;
  now
    assume X '||' Y;
    then [a,A] in Inc_of_Dir(AS) by A1,A2,A3,A4,Def12;
    hence a on A by INCSP_1:def 1;
  end;
  hence thesis by A5;
end;
