reserve x for set;
reserve a,b,c,d,e,r1,r2,r3,r4,r5,r6 for Real;
reserve V for RealLinearSpace;
reserve u,v,v1,v2,v3,w,w1,w2,w3 for VECTOR of V;
reserve W,W1,W2 for Subspace of V;

theorem Th28:
  {u,w,v} is linearly-independent & u <> v & u <> w & v <> w & a
  <> 0 & b <> 0 implies {u,a * w,b * v} is linearly-independent
proof
  assume that
A1: {u,w,v} is linearly-independent & u <> v & u <> w & v <> w and
A2: a <> 0 & b <> 0;
  now
    let c,d,e;
    assume c * u + d * (a * w) + e * (b * v) = 0.V;
    then
A3: 0.V = c * u + d * a * w + e * (b * v) by RLVECT_1:def 7
      .= c * u + d * a * w + e * b * v by RLVECT_1:def 7;
    then d * a = 0 & e * b = 0 by A1,Th7;
    hence c = 0 & d = 0 & e = 0 by A1,A2,A3,Th7,XCMPLX_1:6;
  end;
  hence thesis by Th7;
end;
