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: dom p =Seg len p by FINSEQ_1:def 3;
A2: Seg len ((1_K)*p)=Seg len p & dom ((1_K)*p)=Seg len ((1_K)*p) by
FINSEQ_1:def 3,MATRIXR1:16;
  for i being Nat st i in dom p holds ((1_K)*p).i=p.i
  proof
    let i be Nat;
A3: rng p c= the carrier of K by FINSEQ_1:def 4;
    assume
A4: i in dom p;
    then
A5: p.i in dom((the multF of K)[;]((1_K),id the carrier of K)) by A2,A1,
FUNCT_1:11;
    p.i in rng p by A4,FUNCT_1:3;
    then reconsider b=p.i as Element of K by A3;
    ((1_K)*p).i=((1_K) multfield).(p.i) by A4,FUNCT_1:13
      .=(the multF of K).((1_K),(id the carrier of K).(p.i)) by A5,FUNCOP_1:32
      .=(1_K)*b
      .=p.i;
    hence thesis;
  end;
  hence thesis by A2,A1,FINSEQ_1:13;
end;
