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 Th42:
  for x being POINT of IncProjSp_of(AS),X being Element of the
Lines of ProjHorizon(AS) st [x,[X,2]] in the Inc of IncProjSp_of(AS) holds x is
  Element of the Points of ProjHorizon(AS)
proof
  let x be POINT of IncProjSp_of(AS), X be Element of the Lines of ProjHorizon
  (AS) such that
A1: [x,[X,2]] in the Inc of IncProjSp_of(AS);
  reconsider A=[X,2] as LINE of IncProjSp_of(AS) by Th25;
A2: ex Y st X=PDir(Y) & Y is being_plane by Th15;
  not x is Element of AS
  proof
    assume not thesis;
    then reconsider a=x as Element of AS;
A3: a=x;
    x on A by A1,INCSP_1:def 1;
    hence contradiction by A2,A3,Th27;
  end;
  then ex Y9 st x=LDir(Y9) & Y9 is being_line by Th20;
  hence thesis by Th14;
end;
