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 * u + b * w) + c * (v + u) by RLVECT_1:def 5
      .= a * u + b * u + b * w + c * (v + u) by RLVECT_1:def 3
      .= (a + b) * u + b * w + c * (v + u) by RLVECT_1:def 6
      .= (a + b) * u + b * w + (c * u + c * v) by RLVECT_1:def 5
      .= (a + b) * u + (b * w + (c * u + c * v)) by RLVECT_1:def 3
      .= (a + b) * u + (c * u + (b * w + c * v)) by RLVECT_1:def 3
      .= (a + b) * u + c * u + (b * w + c * v) by RLVECT_1:def 3
      .= (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;
