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 Th31:
  dom T1 = dom T2 & (for p st p in dom T1 holds T1.p = T2.p) implies T1 = T2
proof
  assume that
A1: dom T1 = dom T2 and
A2: for p st p in dom T1 holds T1.p = T2.p;
 now
    let x be object;
    assume x in dom T1;
    then reconsider p = x as Element of dom T1;
 T1.p = T2.p by A2;
    hence T1.x = T2.x;
  end;
  hence thesis by A1;
end;
