reserve n,m,k for Nat;
reserve r,r1 for Real;
reserve f,seq,seq1 for Real_Sequence;
reserve x,y for set;
reserve e1,e2 for ExtReal;
reserve Nseq for increasing sequence of NAT;
reserve v for FinSequence of REAL,
  r,s for Real,
  n,m,i,j,k for Nat;

theorem
  for f being FinSequence,n,m,k st len f = m+1 & n in dom f & k in Seg m
  holds Del(f,n).k = f.k or Del(f,n).k = f.(k+1)
proof
  let f be FinSequence,n,m,k such that
A1: len f = m+1 & n in dom f and
A2: k in Seg m;
  set X = dom f \ {n};
A3: dom f=Seg len f by FINSEQ_1:def 3;
  then
A4: n<=k & k<=m implies Sgm(X).k=k+1 by A1,A2,FINSEQ_3:108;
  rng Sgm(X) = X by FINSEQ_1:def 14;
  then
A5: dom (f*Sgm(X)) = dom Sgm(X) by RELAT_1:27,XBOOLE_1:36;
A6: dom Sgm(X)=Seg len Sgm(X) & Del(f,n) = f*Sgm(X) by FINSEQ_1:def 3
,FINSEQ_3:def 2;
A7: len Sgm(X) = m by A1,A3,FINSEQ_3:107;
  1<=k & k<n implies Sgm(X).k=k by A1,A2,A3,FINSEQ_3:108;
  hence thesis by A2,A6,A5,A7,A4,FINSEQ_3:25,FUNCT_1:12;
end;
