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;

theorem
  rng s c= dom h & s1 is subsequence of s implies h/*s1 is subsequence of h/*s
proof
  assume that
A1: rng s c= dom h and
A2: s1 is subsequence of s;
  consider N such that
A3: s1=s*N by A2,Def13;
  take N;
  thus thesis by A1,A3,FUNCT_2:110;
end;
