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 Th13:
  for f1,i1,i2 st 1<=i2 & i2<=i1 & i1<=len f1 & 1<=j & j<=i1-'i2+1 holds
    mid(f1,i1,i2).j=mid(f1,i2,i1).(i1-i2+1-j+1) & i1-i2+1-j+1=i1-'i2+1-'j+1
proof
  let f1,i1,i2;
  assume that
A1: 1<=i2 and
A2: i2<=i1 and
A3: i1<=len f1 and
A4: 1<=j and
A5: j<=i1-'i2+1;
A6: 1<=i1 by A1,A2,XXREAL_0:2;
  i2-1>=0 by A1,XREAL_1:48;
  then
A7: i1-(i2-1)<=i1-0 by XREAL_1:10;
A8: i2<=len f1 by A2,A3,XXREAL_0:2;
  1-j<=j-j by A4,XREAL_1:9;
  then
A9: i1-'i2+(1-j)<=i1-'i2+0 by XREAL_1:7;
  j-j<=i1-'i2+1-j by A5,XREAL_1:9;
  then i1-'i2+1-'j<=i1-'i2 by A9,XREAL_0:def 2;
  then
A10: i1-'i2+1-'j+1<=i1-'i2+1 by XREAL_1:6;
A11: 1<=(i1-'i2+1-'j+1) by NAT_1:11;
A12: (i1-'i2+1-'j+1) =i1-'i2+1-j+1 by A5,XREAL_1:233
    .=i1-i2+1-j+1 by A2,XREAL_1:233;
A13: i1-'i2+1=i1-i2+1 by A2,XREAL_1:233;
  now
    per cases by A2,XXREAL_0:1;
    case
A14:  i1>i2;
      len mid(f1,i2,i1)=i1-'i2+1 by A1,A2,A3,A6,A8,Th117;
      then
A15:  mid(f1,i2,i1).(i1-'i2+1-'j+1)=f1.((i1-'i2+1-'j+1)+i2-'1) by A1,A2,A3,A6
,A8,A10,A11,Th117;
A16:  len mid(f1,i1,i2)=i1-'i2+1 by A1,A3,A6,A8,A14,Th117;
      (i1-'i2+1-'j+1)+i2-'1 =(i1-'i2+1-'j+1)+i2-1 by A1,NAT_D:37
        .=i1-j+1 by A12
        .=i1-'j+1 by A5,A13,A7,XREAL_1:233,XXREAL_0:2;
      hence thesis by A1,A3,A4,A5,A6,A8,A12,A14,A15,A16,Th117;
    end;
    case
A17:  i1=i2;
      then i1-'i2+1=0+1 by XREAL_1:232
        .=1;
      then i1-i2+1-j+1=0+1-1+1 by A4,A5,A17,XXREAL_0:1
        .=1;
      hence thesis by A12,A17;
    end;
  end;
  hence thesis;
end;
