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 Th19:
  {u,v} is linearly-independent & u <> v & a <> 0 implies {u,a * v
  } is linearly-independent
proof
  assume that
A1: {u,v} is linearly-independent & u <> v and
A2: a <> 0;
  now
    let b,c;
    assume b * u + c * (a * v) = 0.V;
    then
A3: 0.V = b * u + c * a * v by RLVECT_1:def 7;
    then c * a = 0 by A1,RLVECT_3:13;
    hence b = 0 & c = 0 by A1,A2,A3,RLVECT_3:13,XCMPLX_1:6;
  end;
  hence thesis by RLVECT_3:13;
end;
