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);

theorem Th28:
  a=LDir(Y) & [X,1]=A & Y is being_line & X is being_line implies
  (a on A iff Y '||' X)
proof
  assume that
A1: a=LDir(Y) and
A2: [X,1]=A and
A3: Y is being_line and
A4: X is being_line;
A5: now
A6: now
      given K such that
A7:   K is being_line and
A8:   [X,1]=[K,1] and
A9:   LDir(Y) in the carrier of AS & LDir(Y) in K or LDir(Y) = LDir(K );
A10:  K in AfLines(AS) by A7;
A11:  X=K by A8,XTUPLE_0:1;
A12:  now
        assume LDir(Y)=LDir(K);
        then
A13:    Y in Class(LinesParallelity(AS),K) by A10,EQREL_1:23;
        LDir(K)=Class(LinesParallelity(AS),K);
        then consider K9 being Subset of AS such that
A14:    Y=K9 and
A15:    K9 is being_line and
A16:    K '||' K9 by A7,A13,Th9;
        K // K9 by A7,A15,A16,AFF_4:40;
        hence Y '||' X by A7,A11,A14,A15,AFF_4:40;
      end;
      now
        assume that
A17:    LDir(Y) in the carrier of AS and
        LDir(Y) in K;
        a in Dir_of_Lines(AS) by A1,A3,Th14;
        then (the carrier of AS) /\ Dir_of_Lines(AS) <> {} by A1,A17,
XBOOLE_0:def 4;
        then (the carrier of AS) meets Dir_of_Lines(AS) by XBOOLE_0:def 7;
        hence contradiction by Th16;
      end;
      hence Y '||' X by A9,A12;
    end;
    assume a on A;
    then
A18: [a,A] in the Inc of IncProjSp_of(AS) by INCSP_1:def 1;
    not ex K,X9 st K is being_line & X9 is being_plane & LDir(Y)=LDir(K)
    & [X,1]=[PDir(X9),2] & K '||' X9 by XTUPLE_0:1;
    hence Y '||' X by A1,A2,A18,A6,Def11;
  end;
  now
    assume Y '||' X;
    then
A19: X in LDir(Y) by A3,A4,Th9;
A20: LDir(X)=Class(LinesParallelity(AS),X);
    Y in AfLines(AS) by A3;
    then Class(LinesParallelity(AS),X)=Class(LinesParallelity(AS),Y) by A19,
EQREL_1:23;
    then [a,A] in Proj_Inc(AS) by A1,A2,A4,A20,Def11;
    hence a on A by INCSP_1:def 1;
  end;
  hence thesis by A5;
end;
