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;

theorem
  for f being FinSequence,k,i being Nat st
  1<=i & i<=k & k<=len f holds mid(f,1,k).i=f.i
proof
  let f be FinSequence,k,i be Nat;
  assume that
A1: 1<=i and
A2: i<=k and
A3: k<=len f;
A4: i<=k-1+1 by A2;
  1<=k by A1,A2,XXREAL_0:2;
  then mid(f,1,k).i=f.(i+1-'1) by A1,A3,A4,Th121;
  hence thesis by NAT_D:34;
end;
