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 & w + v - v = w & v - v + w = w & w - v + v = w & v + (w
  - v) = w & w + (v - v) = w & v - (v - w) = w
proof
  thus v + w - v = w by RLSUB_2:61;
  thus w + v - v = w by RLSUB_2:61;
  thus
A1: v - v + w = 0.V + w by RLVECT_1:15
    .= w by RLVECT_1:4;
  thus w - v + v = w by RLSUB_2:61;
  thus thesis by A1,RLSUB_2:61,RLVECT_1:29;
end;
