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 Th9:
  X is being_line implies for x holds x in LDir(X) iff ex Y st x=Y
  & Y is being_line & X '||' Y
proof
  assume
A1: X is being_line;
  let x;
A2: now
    assume x in LDir(X);
    then [x,X] in LinesParallelity(AS) by EQREL_1:19;
    then consider K,M such that
A3: [x,X]=[K,M] and
A4: K is being_line and
A5: M is being_line and
A6: K '||' M;
    take Y=K;
A7: X=M by A3,XTUPLE_0:1;
    K // M by A4,A5,A6,AFF_4:40;
    hence x=Y & Y is being_line & X '||' Y by A3,A4,A5,A7,AFF_4:40,XTUPLE_0:1;
  end;
  now
    given Y such that
A8: x=Y and
A9: Y is being_line and
A10: X '||' Y;
    X // Y by A1,A9,A10,AFF_4:40;
    then Y '||' X by A1,A9,AFF_4:40;
    then
    [Y,X] in {[K,M]: K is being_line & M is being_line & K '||' M} by A1,A9;
    hence x in LDir(X) by A8,EQREL_1:19;
  end;
  hence thesis by A2;
end;
