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 p being FinSequence, k1,k2 being Nat
  st k1 < k2 & k1 in dom p holds mid(p,k1,k2) = <*p.k1*> ^ mid(p,k1+1,k2)
proof
  let p be FinSequence, k1,k2 be Nat;
  assume A1: k1 < k2 & k1 in dom p;
  then reconsider D = rng p as non empty set by FUNCT_1:3;
  reconsider q = p as FinSequence of D by FINSEQ_1:def 4;
  mid(q,k1,k2) = <*q.k1*> ^ mid(q,k1+1,k2) by A1,Th126;
  hence thesis;
end;
