reserve i,j,k,l,m,n for Nat,
  D for non empty set,
  f for FinSequence of D;

theorem Th4:
  i > j & i in dom f & j in dom f & k in dom mid(f,i,j) implies (
  mid(f,i,j))/.k = f/.(i-' k +1)
proof
  assume that
A1: i > j and
A2: i in dom f and
A3: j in dom f and
A4: k in dom mid(f,i,j);
A5: 1 <= i & i <= len f by A2,FINSEQ_3:25;
A6: 1 <= k & k <= len mid(f,i,j) by A4,FINSEQ_3:25;
A7: 1 <= j & j <= len f by A3,FINSEQ_3:25;
  thus (mid(f,i,j))/.k = mid(f,i,j).k by A4,PARTFUN1:def 6
    .= f.(i -' k +1) by A1,A5,A7,A6,FINSEQ_6:118
    .= f/.(i-' k +1) by A1,A2,A3,A4,Th2,PARTFUN1:def 6;
end;
