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 Th37:
  for a being Element of the Points of ProjHorizon(AS),a9 being
  Element of the Points of IncProjSp_of(AS),A being Element of the Lines of
  ProjHorizon(AS),A9 being LINE of IncProjSp_of(AS) st a9=a & A9=[A,2] holds (a
  on A iff a9 on A9)
proof
  let a be Element of the Points of ProjHorizon(AS),a9 be Element of the
  Points of IncProjSp_of(AS),A be Element of the Lines of ProjHorizon(AS),A9 be
  LINE of IncProjSp_of(AS) such that
A1: a9=a and
A2: A9=[A,2];
  consider X such that
A3: a=LDir(X) and
A4: X is being_line by Th14;
  consider Y such that
A5: A=PDir(Y) and
A6: Y is being_plane by Th15;
A7: now
    assume a9 on A9;
    then X '||' Y by A1,A2,A3,A4,A5,A6,Th29;
    hence a on A by A3,A4,A5,A6,Th36;
  end;
  now
    assume a on A;
    then X '||' Y by A3,A4,A5,A6,Th36;
    hence a9 on A9 by A1,A2,A3,A4,A5,A6,Th29;
  end;
  hence thesis by A7;
end;
