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
  w <> 0.V & {v,w} is linearly-dependent implies ex a st v = a * w
proof
  assume that
A1: w <> 0.V and
A2: {v,w} is linearly-dependent;
  consider a,b such that
A3: a * v + b * w = 0.V and
A4: a <> 0 or b <> 0 by A2,RLVECT_3:13;
A5: a * v = - b * w by A3,RLVECT_1:6;
  now
    per cases;
    suppose
A6:   a <> 0;
      a" * a * v = a" * (- b * w) by A5,RLVECT_1:def 7;
      then 1 * v = a" * (- b * w) by A6,XCMPLX_0:def 7;
      then v = a" * (- b * w) by RLVECT_1:def 8
        .= a" * ((- b) * w) by Th3
        .= a" * (- b) * w by RLVECT_1:def 7;
      hence thesis;
    end;
    suppose
A7:   a = 0;
      then 0.V = - b * w by A5,RLVECT_1:10;
      then
A8:   0.V = (- b) * w by Th3;
      - b <> 0 by A4,A7;
      hence thesis by A1,A8,RLVECT_1:11;
    end;
  end;
  hence thesis;
end;
