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
  v <> w & {v,w} is linearly-independent & {u,v,w} is linearly-dependent
  implies ex a,b st u = a * v + b * w
proof
  assume that
A1: v <> w & {v,w} is linearly-independent and
A2: {u,v,w} is linearly-dependent;
  consider a,b,c such that
A3: a * u + b * v + c * w = 0.V and
A4: a <> 0 or b <> 0 or c <> 0 by A2,Th7;
  now
    per cases;
    suppose
A5:   a <> 0;
      a * u + (b * v + c * w) = 0.V by A3,RLVECT_1:def 3;
      then a * u = - (b * v + c * w) by RLVECT_1:6;
      then a" * a * u = a" * (- (b * v + c * w)) by RLVECT_1:def 7;
      then 1 * u = a" * (- (b * v + c * w)) by A5,XCMPLX_0:def 7;
      then u = a" * (- (b * v + c * w)) by RLVECT_1:def 8
        .= a" * ((- 1) * (b * v + c * w)) by RLVECT_1:16
        .= a" * (- 1) * (b * v + c * w) by RLVECT_1:def 7
        .= a" * (- 1) * (b * v) + a" * (- 1) * (c * w) by RLVECT_1:def 5
        .= a" * (- 1) * b * v + a" * (- 1) * (c * w) by RLVECT_1:def 7
        .= a" * (- 1) * b * v + a" * (- 1) * c * w by RLVECT_1:def 7;
      hence thesis;
    end;
    suppose
A6:   a = 0;
      then 0.V = 0.V + b * v + c * w by A3,RLVECT_1:10
        .= b * v + c * w by RLVECT_1:4;
      hence thesis by A1,A4,A6,RLVECT_3:13;
    end;
  end;
  hence thesis;
end;
