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 Th25:
  for i,j,n be Nat st 1<=j & j<=n & i<>j holds (Base_FinSeq(K,n,i) ).j=0.K
proof
  let i,j,n be Nat;
  assume that
A1: 1<=j & j<=n and
A2: i<>j;
A3: j in Seg n by A1,FINSEQ_1:1;
A4: len (n |-> (0.K))=n by CARD_1:def 7;
  len (Replace((n |-> (0.K)),i,(1.K))) = len (n |-> (0.K)) by FINSEQ_7:5
    .= n by CARD_1:def 7;
  hence (Base_FinSeq(K,n,i)).j = (Replace((n |-> (0.K)),i,(1.K)))/.j by A1,
FINSEQ_4:15
    .= ((n |-> (0.K)))/.j by A1,A2,A4,FINSEQ_7:10
    .=((n |-> (0.K))).j by A1,A4,FINSEQ_4:15
    .= 0.K by A3,FINSEQ_2:57;
end;
