reserve a,b,m,x,y,i1,i2,i3,i for Integer,
  k,p,q,n for Nat,
  c,c1,c2 for Element of NAT,
  z for set;
reserve fp,fp1 for FinSequence of NAT,

  b,c,d, n for Element of NAT,
  a for Nat;

theorem Th30:
  for b,n st b in dom fp & len fp = n+1 holds Del(fp^<*d*>,b) = Del(fp,b)^<*d*>
proof
  let b,n such that
A1: b in dom fp and
A2: len fp = n+1;
A3: len Del(fp,b) = n by A1,A2,FINSEQ_3:109;
  then
A4: len (Del(fp,b)^<*d*>) = n+1 by FINSEQ_2:16;
A5: len (fp^<*d*>) = (n+1)+1 & b in dom (fp^<*d*>) by A1,A2,FINSEQ_2:16
,FINSEQ_3:22;
  then
A6: len Del(fp^<*d*>,b) = len (Del(fp,b)^<*d*>) by A4,FINSEQ_3:109;
  for k be Nat st 1 <= k & k <= len Del(fp^<*d*>,b) holds Del(fp^<*d*>,b).
  k=(Del(fp,b)^<*d*>).k
  proof
    let k be Nat such that
A7: 1 <=k and
A8: k <= len Del(fp^<*d*>,b);
A9: k in dom fp by A2,A4,A6,A7,A8,FINSEQ_3:25;
    per cases;
    suppose
A10:  k<b;
      b <= n+1 by A1,A2,FINSEQ_3:25;
      then k < n+1 by A10,XXREAL_0:2;
      then k <= n by NAT_1:13;
      then k in dom Del(fp,b) by A3,A7,FINSEQ_3:25;
      then
A11:  (Del(fp,b)^<*d*>).k = Del(fp,b).k by FINSEQ_1:def 7
        .= fp.k by A10,FINSEQ_3:110;
      set fpd = fp^<*d*>;
      Del(fpd,b).k = fpd.k by A10,FINSEQ_3:110
        .= fp.k by A9,FINSEQ_1:def 7;
      hence thesis by A11;
    end;
    suppose
A12:  b<=k;
      then
A13:  Del(fp^<*d*>,b).k = (fp^<*d*>).(k+1) by A4,A5,A6,A8,FINSEQ_3:111;
      per cases by A4,A6,A8,NAT_1:8;
      suppose
A14:    k<=n;
        then k in dom Del(fp,b) by A3,A7,FINSEQ_3:25;
        then
A15:    (Del(fp,b)^<*d*>).k = Del(fp,b).k by FINSEQ_1:def 7
          .= fp.(k+1) by A1,A2,A12,A14,FINSEQ_3:111;
A16:    k+1 >= 1 by NAT_1:11;
        k+1 <= n+1 by A14,XREAL_1:6;
        then (k+1) in dom fp by A2,A16,FINSEQ_3:25;
        hence thesis by A13,A15,FINSEQ_1:def 7;
      end;
      suppose
A17:    k=n+1;
        then Del(fp^<*d*>,b).k = d by A2,A13,FINSEQ_1:42;
        hence thesis by A3,A17,FINSEQ_1:42;
      end;
    end;
  end;
  hence thesis by A4,A5,FINSEQ_3:109;
end;
