reserve V for RealLinearSpace;
reserve p,q,u,v,w,y for VECTOR of V;
reserve a,b,c,d for Real;
reserve AS for non empty AffinStruct;
reserve a,b,c,d for Element of AS;
reserve x,z for object;

theorem Th22:
  [[u,v],[w,y]] in DirPar(V) iff u,v // w,y
proof
  thus [[u,v],[w,y]] in DirPar(V) implies u,v // w,y
  proof
    assume [[u,v],[w,y]] in DirPar(V);
    then consider u9,v9,w9,y9 being VECTOR of V such that
A1: [u,v]=[u9,v9] and
A2: [w,y]=[w9,y9] and
A3: u9,v9 // w9,y9 by Def3;
A4: w = w9 by A2,XTUPLE_0:1;
    u = u9 & v = v9 by A1,XTUPLE_0:1;
    hence thesis by A2,A3,A4,XTUPLE_0:1;
  end;
  thus thesis by Def3;
end;
