reserve p,q,r for FinSequence;
reserve u,v,x,y,y1,y2,z for object, A,D,X,Y for set;
reserve i,j,k,l,m,n for Nat;
reserve J for Nat;

theorem
  for f being FinSequence st len f = m+1 & n in dom f & n<=k & k<=m
  holds Del(f,n).k = f.(k+1)
proof
  let f be FinSequence;
  assume that
A1: len f = m+1 and
A2: n in dom f and
A3: n<=k and
A4: k<=m;
  set X = dom f \ {n};
A5: dom f=Seg len f by FINSEQ_1:def 3;
  rng Sgm(X) = X by FINSEQ_1:def 14;
  then
A6: dom (f*Sgm(X)) = dom Sgm(X) by RELAT_1:27,XBOOLE_1:36;
A7: len Sgm(X) = m by A1,A2,A5,Th105;
A8: dom Sgm(X)=Seg len Sgm(X) by FINSEQ_1:def 3;
  1<=n by A2,Th25;
  then 1<=k by A3,XXREAL_0:2;
  then
A9: k in Seg m by A4,FINSEQ_1:1;
  then n<=k & k<=m implies Sgm(X).k = k+1 by A1,A2,A5,Th106;
  hence thesis by A3,A4,A9,A8,A6,A7,FUNCT_1:12;
end;
