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 k < n holds Del(f,n).k = f.k
proof
  let f be FinSequence;
  assume
A1: k<n;
  per cases;
  suppose that
A2: n in dom f and
A3: f <> {};
    consider m being Nat such that
A4: len f = m+1 by A3,NAT_1:6;
    now
      per cases;
      suppose
A5:     1 <= k;
        set X = dom f \ {n};
A6:     dom Sgm(X)=Seg len Sgm(X) by FINSEQ_1:def 3;
A7:     dom f=Seg len f by FINSEQ_1:def 3;
        then
A8:     len Sgm X = m by A4,A2,Th105;
        rng Sgm(X) = X by FINSEQ_1:def 14;
        then
A9:     dom (f*Sgm X) = dom Sgm X by RELAT_1:27,XBOOLE_1:36;
        n<=m+1 by A4,A2,Th25;
        then k<m+1 by A1,XXREAL_0:2;
        then k<=m by NAT_1:13;
        then
A10:     k in Seg m by A5,FINSEQ_1:1;
        then 1<=k & k<n implies Sgm(X).k = k by A4,A2,A7,Th106;
        hence thesis by A1,A5,A10,A6,A9,A8,FUNCT_1:12;
      end;
      suppose
A11:    not 1 <= k;
        Seg len Del (f,n) = Seg m by A4,A2,Th107;
        then dom Del (f,n) = Seg m by FINSEQ_1:def 3;
        then
A12:    not k in dom Del (f,n) by A11,FINSEQ_1:1;
        not k in dom f by A11,Th25;
        then f.k = {} by FUNCT_1:def 2;
        hence thesis by A12,FUNCT_1:def 2;
      end;
    end;
    hence thesis;
  end;
  suppose
    not n in dom f or f = {};
    hence thesis by Th102;
  end;
end;
