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 Th20:
  for x holds (x is POINT of IncProjSp_of(AS) iff (x is Element of
  AS or ex X st x=LDir(X) & X is being_line))
proof
  let x;
A1: now
A2: now
      given X such that
A3:   x=LDir(X) and
A4:   X is being_line;
      x in Dir_of_Lines( AS ) by A3,A4,Th14;
      hence x is POINT of IncProjSp_of(AS) by XBOOLE_0:def 3;
    end;
    assume x is Element of AS or ex X st x=LDir(X) & X is being_line;
    hence x is POINT of IncProjSp_of(AS) by A2,XBOOLE_0:def 3;
  end;
  now
    assume
A5: x is POINT of IncProjSp_of(AS);
    x in Dir_of_Lines(AS) implies ex X st x=LDir(X) & X is being_line by Th14;
    hence x is Element of AS or ex X st x=LDir(X) & X is being_line by A5,
XBOOLE_0:def 3;
  end;
  hence thesis by A1;
end;
