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 Th10:
  for f1,i1,i2,j st 1<=i1 & i1<=i2 & i2<=len f1 holds
    mid(f1,i1,i2).(len mid(f1,i1,i2))=f1.i2
proof
  let f1,i1,i2,j;
  assume that
A1: 1<=i1 and
A2: i1<=i2 and
A3: i2<=len f1;
A4: i1<=len f1 by A2,A3,XXREAL_0:2;
A5: 1<=i2 by A1,A2,XXREAL_0:2;
  then len mid(f1,i1,i2)=i2-'i1+1 by A1,A2,A3,A4,Th117;
  then 1<=len mid(f1,i1,i2) by NAT_1:11;
  then
A6: mid(f1,i1,i2).(len mid(f1,i1,i2)) =f1.(len mid(f1,i1,i2)+i1-'1) by A1,A2,A3
,A5,A4,Th117
    .=f1.(i2-'i1+1+i1-'1) by A1,A2,A3,A5,A4,Th117;
  i2-'i1+1+i1=i2-i1+1+i1 by A2,XREAL_1:233
    .=i2+1;
  hence thesis by A6,NAT_D:34;
end;
