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 Th11:
  X is being_line & Y is being_line implies (LDir(X)=LDir(Y) iff X // Y)
proof
  assume that
A1: X is being_line and
A2: Y is being_line;
A3: LDir(Y)= Class(LinesParallelity(AS),Y);
A4: Y in AfLines(AS) by A2;
A5: now
    assume LDir(X)=LDir(Y);
    then X in Class(LinesParallelity(AS),Y) by A4,EQREL_1:23;
    then ex Y9 st X=Y9 & Y9 is being_line & Y '||' Y9 by A2,A3,Th9;
    hence X // Y by A2,AFF_4:40;
  end;
A6: LDir(X)=Class(LinesParallelity(AS),X);
A7: X in AfLines(AS) by A1;
  now
    assume X // Y;
    then X '||' Y by A1,A2,AFF_4:40;
    then Y in Class(LinesParallelity(AS),X) by A1,A2,A6,Th9;
    hence LDir(X)=LDir(Y) by A7,EQREL_1:23;
  end;
  hence thesis by A5;
end;
