reserve X for Banach_Algebra,
  w,z,z1,z2 for Element of X,
  k,l,m,n,n1,n2 for Nat,
  seq,seq1,seq2,s,s9 for sequence of X,
  rseq for Real_Sequence;

theorem Th13:
  (for k st 0 < k holds ((k-'1)! ) * k = k!) & for m,k st k <= m
  holds ((m-'k)! ) * (m+1-k) = (m+1-'k)!
proof
A1: now
    let k;
A2: k in NAT by ORDINAL1:def 12;
    assume 0 < k;
    then 0+1 <= k by INT_1:7,A2;
    then k-'1+1=k-1+1 by XREAL_1:233
      .=k;
    hence k! =(k-'1)! *k by NEWTON:15;
  end;
  now
    let m,k such that
A3: k <= m;
    m <= m+1 by XREAL_1:29;
    then m+1-'k=m+1-k by A3,XREAL_1:233,XXREAL_0:2
      .=m-k+1
      .=m-'k+1 by A3,XREAL_1:233;
    hence (m+1-'k)!=((m-'k)! ) *( (m-'k)+1) by NEWTON:15
      .=((m-'k)! ) *(m-k+1) by A3,XREAL_1:233
      .=((m-'k)! ) *(m+1-k);
  end;
  hence thesis by A1;
end;
