reserve x,y for object,X for set,
  f for Function,
  R,S for Relation;
reserve e1,e2 for ExtReal;
reserve s,s1,s2,s3 for sequence of X;
reserve XX for non empty set,
        ss,ss1,ss2,ss3 for sequence of XX;
reserve X,Y for non empty set,
  Z for set;
reserve s,s1 for sequence of X,
  h,h1 for PartFunc of X,Y,
  h2 for PartFunc of Y ,Z,
  x for Element of X,
  N for increasing sequence of NAT;
reserve i,j for Nat;
reserve n for Nat;

theorem Th27:
  rng s c= dom h implies (h/*s)^\n=h/*(s^\n)
proof
  assume
A1: rng s c= dom h;
  let m be Element of NAT;
A2: rng (s^\n) c= rng s by Th21;
   reconsider mn = m+n as Element of NAT by ORDINAL1:def 12;
  thus ((h/*s)^\n).m = (h/*s).(m+n) by NAT_1:def 3
    .= h.(s.(mn)) by A1,FUNCT_2:108
    .= h.((s^\n).m) by NAT_1:def 3
    .= (h/*(s^\n)).m by A1,A2,FUNCT_2:108,XBOOLE_1:1;
end;
