reserve i, j, m, n for Nat,
  z, B0 for set,
  f, x0 for real-valued FinSequence;

theorem Th29:
  for x being Element of REAL n st 1<=i & i<=n holds
    |( x,Base_FinSeq(n,i) )| = x.i
proof
  let x be Element of REAL n;
  assume that
A1: 1<=i and
A2: i<=n;
  set x2=Base_FinSeq(n,i);
A3: len x=n by CARD_1:def 7;
A4: len x2=n by MATRIXR2:74;
  then
A5: len (mlt(x,x2))=n by A3,MATRPROB:30;
A6: for j being Nat st 1<=j & j<=n holds (mlt(x,x2)).j=((x/.i)*(Base_FinSeq(
  n,i))).j
  proof
    let j be Nat;
    assume that
A7: 1<=j and
A8: j<=n;
    reconsider j0=j as Element of NAT by ORDINAL1:def 12;
A9: now
      per cases;
      case i=j;
        hence ((x/.i)*(Base_FinSeq(n,i))).j=(x/.j)*((Base_FinSeq(n,i)).j) by
RVSUM_1:44;
      end;
      case i<>j;
        then
A10:    (Base_FinSeq(n,i)).j0=0 by A7,A8,MATRIXR2:76;
        ((x/.i)*(Base_FinSeq(n,i))).j=(x/.i)*((Base_FinSeq(n,i)).j) by
RVSUM_1:44
          .= (x/.j)*((Base_FinSeq(n,i)).j) by A10;
        hence ((x/.i)*(Base_FinSeq(n,i))).j=(x/.j)*((Base_FinSeq(n,i)).j);
      end;
    end;
    (mlt(x,x2)).j=(x.j)*(x2.j) by RVSUM_1:59;
    hence (mlt(x,x2)).j=((x/.i)*(Base_FinSeq(n,i))).j by A3,A7,A8,A9,
FINSEQ_4:15;
  end;
  len ((x/.i)*(Base_FinSeq(n,i)))=n by A4,RVSUM_1:117;
  then (mlt(x,x2))= (x/.i)*(Base_FinSeq(n,i)) by A5,A6,FINSEQ_1:14;
  then Sum mlt(x,x2)=(x/.i)*(Sum (Base_FinSeq(n,i))) by RVSUM_1:87
    .=(x/.i)*1 by A1,A2,Th26
    .=x.i by A1,A2,A3,FINSEQ_4:15;
  hence |( x, Base_FinSeq(n,i) )| = x.i;
end;
