reserve x,y,z,a,b,c,X,X1,X2,Y,Z for set,
  W,W1,W2 for Tree,
  w,w9 for Element of W,
  f for Function,
  D,D9 for non empty set,
  i,k,k1,k2,l,m,n for Nat,
  v,v1,v2 for FinSequence,
  p,q,r,r1,r2 for FinSequence of NAT;
reserve C for Chain of W,
  B for Branch of W;
reserve T,T1,T2 for DecoratedTree;

theorem Th32:
  p in dom T implies rng (T|p) c= rng T
proof
  assume
A1: p in dom T;
  let x be object;
  assume x in rng (T|p);
  then consider y being object such that
A2: y in dom (T|p) and
A3: x = (T|p).y by FUNCT_1:def 3;
  reconsider y as Element of dom (T|p) by A2;
A4: dom (T|p) = (dom T)|p by Def10;
then A5: p^y in dom T by A1,TREES_1:def 6;
 x = T.(p^y) by A3,A4,Def10;
  hence thesis by A5,FUNCT_1:def 3;
end;
