reserve T1,T2,T3 for TopSpace,
  A1 for Subset of T1, A2 for Subset of T2, A3 for Subset of T3;
reserve n,k for Nat;
reserve M,N for non empty TopSpace;
reserve p,q,p1,p2 for Point of TOP-REAL n;
reserve r for Real;

theorem Th28:
  Base_FinSeq(n+1,n+1) = (0.TOP-REAL n) ^ <* 1 *>
proof
  set N = Base_FinSeq(n+1,n+1);
  set p = 0.TOP-REAL n;
  set q = <* 1 *>;
A1: dom N = Seg (n + 1) by FINSEQ_1:89
  .= Seg (n + len q) by FINSEQ_1:39
  .= Seg (len p + len q) by CARD_1:def 7;
A2: for k st k in dom p holds N.k = p.k
  proof
    let k;
    assume k in dom p;
    then
 k in Seg n by FINSEQ_1:89;
    then
A3: 1 <= k & k <= n by FINSEQ_1:1;
    0+n <= 1+n by XREAL_1:6;
    then
A4: k <= n+1 by A3,XXREAL_0:2;
A5: n+1 <> k by A3,NAT_1:13;
    thus N.k = 0 by A4,A3,A5,MATRIXR2:76
    .= (0*n).k
    .= p.k by EUCLID:70;
  end;
  for k st k in dom q holds N.(len p + k) = q.k
  proof
    let k;
    assume k in dom q;
    then k in {1} by FINSEQ_1:2,38;
    then A6: k = 1 by TARSKI:def 1;
    A7: 0+1 <= n+1 by XREAL_1:6;
    thus N.(len p + k) = Base_FinSeq(n+1,n+1).(n+1) by A6,CARD_1:def 7
    .= 1 by A7,MATRIXR2:75
    .= q.k by A6;
  end;
  hence thesis by A1,A2,FINSEQ_1:def 7;
end;
