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
  {u,w,v} is linearly-independent & u <> v & u <> w & v <> w implies {u,
  w - u,v - u} is linearly-independent
proof
  assume
A1: {u,w,v} is linearly-independent & u <> v & u <> w & v <> w;
  now
    let a,b,c;
    assume a * u + b * (w - u) + c * (v - u) = 0.V;
    then
A2: 0.V = a * u + (b * w - b * u) + c * (v - u) by RLVECT_1:34
      .= a * u + (b * w + - b * u) + (c * v - c * u) by RLVECT_1:34
      .= a * u + - b * u + b * w + (- c * u + c * v) by RLVECT_1:def 3
      .= a * u + - b * u + (b * w + (- c * u + c * v)) by RLVECT_1:def 3
      .= a * u + - b * u + (- c * u + (b * w + c * v)) by RLVECT_1:def 3
      .= a * u + - b * u + - c * u + (b * w + c * v) by RLVECT_1:def 3
      .= a * u + b * (- u) + - c * u + (b * w + c * v) by RLVECT_1:25
      .= a * u + (- b) * u + - c * u + (b * w + c * v) by RLVECT_1:24
      .= a * u + (- b) * u + c * (- u) + (b * w + c * v) by RLVECT_1:25
      .= a * u + (- b) * u + (- c) * u + (b * w + c * v) by RLVECT_1:24
      .= (a + (- b)) * u + (- c) * u + (b * w + c * v) by RLVECT_1:def 6
      .= (a + (- b) + (- c)) * u + (b * w + c * v) by RLVECT_1:def 6
      .= (a + (- b) + (- c)) * u + b * w + c * v by RLVECT_1:def 3;
    then a + (- b) + (- c) = 0 & b = 0 by A1,Th7;
    hence a = 0 & b = 0 & c = 0 by A1,A2,Th7;
  end;
  hence thesis by Th7;
end;
