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 Th35:
  for x being FinSequence of K st len x=m & 1<=i & i<=m holds |( x
  ,Base_FinSeq(K,m,i) )|=x.i & |( x,Base_FinSeq(K,m,i) )|=x/.i
proof
  let x be FinSequence of K;
  assume that
A1: len x=m and
A2: 1<=i & i<=m;
A3: for j st j<>i & 1<=j & j<=m holds (mlt (x,Base_FinSeq(K,m,i))).j= 0.K
  by A1,A2,Th34;
A4: len (Base_FinSeq(K,m,i))=m by Th23;
A5: rng (Base_FinSeq(K,m,i)) c= the carrier of K by FINSEQ_1:def 4;
  dom (the multF of K)=[:the carrier of K,the carrier of K:] & rng (x) c=
  the carrier of K by FINSEQ_1:def 4,FUNCT_2:def 1;
  then [:rng (x), rng (Base_FinSeq(K,m,i)):] c= dom (the multF of K) by A5,
ZFMISC_1:96;
  then dom ((the multF of K).:(x,Base_FinSeq(K,m,i))) =dom x /\ dom (
  Base_FinSeq(K,m,i)) by FUNCOP_1:69;
  then dom (mlt (x,Base_FinSeq(K,m,i)))= dom x /\ dom (Base_FinSeq(K,m,i)) by
FVSUM_1:def 7
    .= Seg m /\ dom (Base_FinSeq(K,m,i)) by A1,FINSEQ_1:def 3
    .= Seg m /\ Seg m by A4,FINSEQ_1:def 3
    .= Seg m;
  then
A6: len (mlt (x,Base_FinSeq(K,m,i)))=m by FINSEQ_1:def 3;
A7: x/.i=x.i by A1,A2,FINSEQ_4:15;
  then (mlt(x,Base_FinSeq(K,m,i))).i=x/.i by A1,A2,Th34;
  then
A8: Sum (mlt (x,Base_FinSeq(K,m,i)))=x.i by A2,A7,A6,A3,Th32;
  hence |( x,Base_FinSeq(K,m,i) )|=x.i by FVSUM_1:def 9;
  x.i=x/.i by A1,A2,FINSEQ_4:15;
  hence thesis by A8,FVSUM_1:def 9;
end;
