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 Th21:
  a <> b implies {a * v,b * v} is linearly-dependent
proof
  assume
A1: a <> b;
  now
    per cases;
    suppose
      v = 0.V;
      then a * v = 0.V by RLVECT_1:10;
      hence thesis by RLVECT_3:11;
    end;
    suppose
A2:   v <> 0.V;
A3:   b * (a * v) + (- a) * (b * v) = a * b * v + (- a) * (b * v) by
RLVECT_1:def 7
        .= a * b * v - a * (b * v) by Th3
        .= a * b * v - a * b * v by RLVECT_1:def 7
        .= 0.V by RLVECT_1:15;
A4:   not (b = 0 & - a = 0) by A1;
      a * v <> b * v by A1,A2,RLVECT_1:37;
      hence thesis by A3,A4,RLVECT_3:13;
    end;
  end;
  hence thesis;
end;
