reserve q,th,r for Real,
  a,b,p for Real,
  w,z for Complex,
  k,l,m,n,n1,n2 for Nat,
  seq,seq1,seq2,cq1 for Complex_Sequence,
  rseq,rseq1,rseq2 for Real_Sequence,
  rr for set,
  hy1 for 0-convergent non-zero Real_Sequence;

theorem Th2:
  (0 < k implies ((k-'1)! ) * k = k!) &
  (k <= m implies ((m-'k)!) * (m+1-k) = (m+1-'k)!)
proof
A1: now
    let k;
    assume 0 < k;
then  0+1 <= k by INT_1:7;
then  k-'1+1=k-1+1 by XREAL_1:233
      .=k;
    hence k! =(k-'1)! * k by Th1;
  end;
 now
    let m,k such that
A2: k <= m;
 m <= m+1 by XREAL_1:29;
then  m+1-'k=m+1-k by A2,XREAL_1:233,XXREAL_0:2
      .=m-k+1
      .=m-'k+1 by A2,XREAL_1:233;
    hence (m+1-'k)!=((m-'k)! ) * (m-'k+1+0*<i>) by Th1
      .=((m-'k)! ) * (m-k+1+0*<i>) by A2,XREAL_1:233
      .=((m-'k)! ) * (m+1-k);
  end;
  hence thesis by A1;
end;
