reserve i,j,k,n,l for Element of NAT,
  K for Field,
  a,b,c for Element of K,
  p ,q for FinSequence of K,
  M1,M2,M3 for Matrix of n,K;

theorem
  (-1_K)*p=-p
proof
A1: Seg len ((-1_K)*p)=Seg len p & dom ((-1_K)*p)=Seg len ((-1_K)*p) by
FINSEQ_1:def 3,MATRIXR1:16;
A2: dom p =Seg len p by FINSEQ_1:def 3;
A3: for i being Nat st i in dom p holds ((-1_K)*p).i=(-p).i
  proof
    let i be Nat;
A4: rng p c= the carrier of K by FINSEQ_1:def 4;
    assume
A5: i in dom p;
    then
A6: p.i in dom((the multF of K)[;]((-1_K),id the carrier of K)) by A1,A2,
FUNCT_1:11;
    p.i in rng p by A5,FUNCT_1:3;
    then reconsider b=p.i as Element of K by A4;
    (-1_K)*b+b=(-(1_K))*b+(1_K)*b
      .=(-(1_K) + (1_K))*b by VECTSP_1:def 7
      .=(0.K)*b by RLVECT_1:5
      .=0.K;
    then (-1_K)*b+(b+-b)=0.K+-b by RLVECT_1:def 3;
    then 0.K + -b=(-(1_K))*b + 0.K by RLVECT_1:5
      .=(-(1_K))*b by RLVECT_1:4;
    then
A7: (-1_K)*b=-b by RLVECT_1:4;
    ((-1_K)*p).i=((-1_K) multfield).(p.i) by A5,FUNCT_1:13
      .=(the multF of K).((-1_K),(id the carrier of K).(p.i)) by A6,FUNCOP_1:32
      .=(the multF of K).(-1_K,b)
      .=(comp K).b by A7,VECTSP_1:def 13
      .=(-p).i by A5,FUNCT_1:13;
    hence thesis;
  end;
  p is Element of (len p)-tuples_on the carrier of K by FINSEQ_2:92;
  then -p is Element of (len p)-tuples_on the carrier of K by FINSEQ_2:113;
  then
A8: len (-p)=len p by CARD_1:def 7;
  dom (-p)=Seg len (-p) & dom p=Seg len p by FINSEQ_1:def 3;
  hence thesis by A1,A3,A8,FINSEQ_1:13;
end;
