theorem Th27:
  {u,w,v} is linearly-independent & u <> v & u <> w & v <> w & a
  <> 0 implies {u,w,a * v} is linearly-independent
proof
  assume that
A1: {u,w,v} is linearly-independent & u <> v & u <> w & v <> w and
A2: a <> 0;
  now
    let b,c,d;
    assume b * u + c * w + d * (a * v) = 0.V;
    then
A3: 0.V = b * u + c * w + d * a * v by RLVECT_1:def 7;
    then d * a = 0 by A1,Th7;
    hence b = 0 & c = 0 & d = 0 by A1,A2,A3,Th7,XCMPLX_1:6;
  end;
  hence thesis by Th7;
end;
