theorem
  r * A is affinely-independent & r <> 0 implies A is affinely-independent
 proof
  assume that
   A1: r*A is affinely-independent and
   A2: r<>0;
  r"*(r*A)=(r"*r)*A by Th10
   .=1*A by A2,XCMPLX_0:def 7
   .=A by Th11;
  hence thesis by A1;
 end;
