reserve AS for AffinSpace;
reserve a,a9,b,b9,c,d,o,p,q,r,s,x,y,z,t,u,w for Element of AS;
reserve A,C,D,K for Subset of AS;

theorem Th26:
  c in A & d in A & A is being_line & c <>d implies (a,b // A iff a,b // c,d)
proof
  assume that
A1: c in A and
A2: d in A and
A3: A is being_line and
A4: c <>d;
  thus a,b // A implies a,b // c,d
  proof
    assume a,b // A;
    then consider p,q such that
A5: p<>q and
A6: A=Line(p,q) and
A7: a,b // p,q;
    p,q // c,d by A1,A2,A5,A6,Th21;
    hence thesis by A5,A7,Th4;
  end;
    assume
A8: a,b // c,d;
    A=Line(c,d) by A1,A2,A3,A4,Lm6;
    hence thesis by A4,A8;
end;
