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 Th22:
  A is being_line & a in A implies (b in A iff a,b // A)
proof
  assume that
A1: A is being_line and
A2: a in A;
    consider p,q such that
A3: p<>q and
A4: A=Line(p,q) by A1;
    hereby assume b in A;
      then p,q // a,b by A2,A3,A4,Th21;
      then a,b // p,q by Th3;
      hence a,b // A by A3,A4;
    end;
    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 // a,b by A7,Th3;
    hence b in A by A2,A5,A6,Th21;
end;
