reserve x,y,z for set;
reserve f,f1,f2,f3 for FinSequence,
  p,p1,p2,p3 for set,
  i,k for Nat;
reserve D for non empty set,
  p,p1,p2,p3 for Element of D,
  f,f1,f2 for FinSequence of D;
reserve D for non empty set;
reserve p, q for FinSequence,
  X, Y, x, y for set,
  D for non empty set,
  i, j, k, l, m, n, r for Nat;
reserve a, a1, a2 for TwoValued Alternating FinSequence;
reserve fs, fs1, fs2 for FinSequence of X,
  fss, fss2 for Subset of fs;
reserve F, F1 for FinSequence of INT,
  k, m, n, ma for Nat;
reserve i,j,k,m,n for Nat,
  D for non empty set,
  p for Element of D,
  f for FinSequence of D;
reserve D for non empty set,
  p for Element of D,
  f for FinSequence of D;
reserve f for circular FinSequence of D;
reserve i, i1, i2, j, k for Nat;
reserve D for non empty set,
  f1 for FinSequence of D;

theorem Th15:
  for f1 being FinSequence st
  k in dom f1 holds mid(f1,k,k)=<*f1.k*> & len mid(f1,k,k)=1
proof
  let f1 be FinSequence;
  assume
A0: k in dom f1; then
A1: 1<=k by FINSEQ_3:25;
A2: k<=len f1 by A0,FINSEQ_3:25;
  k-'1+1<=len f1 by A1,A2,XREAL_1:235;
  then
A4: k-'1+1-(k-'1)<=len f1-(k-'1) by XREAL_1:9;
  len (f1/^(k-'1))=len f1-'(k-'1) by RFINSEQ:29;
  then
A5: 1<=len (f1/^(k-'1)) by A4,NAT_D:39;
  k-'1+1=k by A1,XREAL_1:235;
  then
A6: (f1/^(k-'1)).1=f1.k by A2,Th113;
  k-'k+1=k-k+1 by XREAL_1:233
    .=1;
  then mid(f1,k,k)=(f1/^(k-'1))|1 by Def3
    .=<*(f1/^(k-'1)).1*> by A5,CARD_1:27,FINSEQ_5:20;
  hence thesis by A6,FINSEQ_1:39;
end;
