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;

theorem Th23:
  for x holds (x is LINE of IncProjSp_of(AS) iff ex X st (x=[X,1]
  & X is being_line or x=[PDir(X),2] & X is being_plane))
proof
  let x;
A1: now
    given X such that
A2: x=[X,1] & X is being_line or x=[PDir(X),2] & X is being_plane;
A3: now
      assume that
A4:   x=[PDir(X),2] and
A5:   X is being_plane;
      x in [:Dir_of_Planes(AS),{2}:] by A4,A5,Th19;
      hence x is LINE of IncProjSp_of(AS) by XBOOLE_0:def 3;
    end;
    now
      assume that
A6:   x=[X,1] and
A7:   X is being_line;
      x in [:AfLines(AS),{1}:] by A6,A7,Th18;
      hence x is LINE of IncProjSp_of(AS) by XBOOLE_0:def 3;
    end;
    hence x is LINE of IncProjSp_of(AS) by A2,A3;
  end;
  now
A8: x in [:Dir_of_Planes(AS),{2}:] implies ex X st x=[X,1] & X is
    being_line or x=[PDir(X),2] & X is being_plane by Th19;
    assume
A9: x is LINE of IncProjSp_of(AS);
    x in [:AfLines(AS),{1}:] implies ex X st x=[X,1] & X is being_line or
    x=[PDir(X),2] & X is being_plane by Th18;
    hence
    ex X st x=[X,1] & X is being_line or x=[PDir(X),2] & X is being_plane
    by A9,A8,XBOOLE_0:def 3;
  end;
  hence thesis by A1;
end;
