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
  for f1,i1,i2 st 1<=i1 & i1<=i2 & i2<=len f1 & 1<=j & j<=i2-'i1+1 holds
  mid(f1,i1,i2).j=mid(f1,i2,i1).(i2-i1+1-j+1) & i2-i1+1-j+1=i2-'i1+1-'j+1
proof
  let f1,i1,i2;
  assume that
A1: 1<=i1 and
A2: i1<=i2 and
A3: i2<=len f1 and
A4: 1<=j and
A5: j<=i2-'i1+1;
  set k=i2-'i1+1-'j+1;
  i2-i1+1-j+1=i2-'i1+1-j+1 by A2,XREAL_1:233
    .=i2-'i1+1-'j+1 by A5,XREAL_1:233;
  then
A6: i2-i1+1-k+1=j;
  j-1>=0 by A4,XREAL_1:48;
  then
A7: k+0<=k+(j-1) by XREAL_1:7;
A8: 1<=k by NAT_1:11;
  i2-'i1+1-j=i2-'i1+1-'j by A5,XREAL_1:233;
  hence thesis by A1,A2,A3,A8,A7,A6,Th13;
end;
