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 Th31:
  a <> b implies {a * v,b * v,w} is linearly-dependent
proof
  assume a <> b;
  then {a * v,b * v} is linearly-dependent by Th21;
  hence thesis by RLVECT_3:5,SETWISEO:2;
end;
