reserve

  k,n,m,i,j for Element of NAT,
  K for Field;
reserve L for non empty addLoopStr;
reserve G for non empty multLoopStr;

theorem Th6:
  for a being Element of K, x being FinSequence of K holds -a*x = (
  -a)*x & -a*x = a*(-x)
proof
  let a be Element of K, x be FinSequence of K;
  set n=len x;
  reconsider x0=x as Element of n-tuples_on (the carrier of K) by Th1;
  reconsider y=a*x0 as Element of (n-tuples_on (the carrier of K));
  thus -a*x=(-1_K)*y by FVSUM_1:59
    .=((-1_K)*a)*x0 by FVSUM_1:54
    .=(- 1_K *a)*x0 by VECTSP_1:8
    .=(- a)*x;
  thus -a*x= (-1_K)*y by FVSUM_1:59
    .=((-1_K)*a)*x0 by FVSUM_1:54
    .=a*((-1_K)*x0) by FVSUM_1:54
    .=a*(-x) by FVSUM_1:59;
end;
