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

theorem Th8:
  i in dom f & j in dom f implies (mid(f,i,j))/.1 = f/.i
proof
  assume
A1: i in dom f;
  then
A2: 1 <= i & i <= len f by FINSEQ_3:25;
  assume
A3: j in dom f;
  then
A4: 1 <= j & j <= len f by FINSEQ_3:25;
  mid(f,i,j) is non empty by A1,A3,Th7;
  then 1 in dom mid(f,i,j) by FINSEQ_5:6;
  hence (mid(f,i,j))/.1 = mid(f,i,j).1 by PARTFUN1:def 6
    .= f.i by A2,A4,FINSEQ_6:118
    .= f/.i by A1,PARTFUN1:def 6;
end;
