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;

theorem Th82:
  p in rng f & p..f > k implies (f/^k):-p = f:-p
proof
  assume that
A1: p in rng f and
A2: p..f > k;
  thus (f/^k):-p =<*p*>^(f/^k/^(p..(f/^k))) by FINSEQ_5:def 2
    .=<*p*>^(f/^(k + p..(f/^k))) by Th81
    .=<*p*>^(f/^p..f) by A1,A2,Lm3
    .= f:-p by FINSEQ_5:def 2;
end;
