reserve x,y,z for object, X,Y for set,
  i,k,n for Nat,
  p,q,r,s for FinSequence,
  w for FinSequence of NAT,
  f for Function;

theorem
  for T,T1,T2 being Tree holds
  tree(T1,T2) with-replacement (<*0*>, T) = tree(T,T2) &
  tree(T1,T2) with-replacement (<*1*>, T) = tree(T1,T)
proof
  let T,T1,T2 be Tree;
A1: len {} = 0;
A2: <*T1*> = {}^<*T1*> by FINSEQ_1:34;
A3: <*T*> = {}^<*T*> by FINSEQ_1:34;
A4: <*T1,T2*>^{} = <*T1,T2*> by FINSEQ_1:34;
A5: <*T1,T*>^{} = <*T1,T*> by FINSEQ_1:34;
A6: len <*T1*> = 1 by FINSEQ_1:40;
A7: <*T1,T2*> = <*T1*>^<*T2*> by FINSEQ_1:def 9;
A8: <*T1,T*> = <*T1*>^<*T*> by FINSEQ_1:def 9;
  <*T,T2*> = <*T*>^<*T2*> by FINSEQ_1:def 9;
  hence thesis by A1,A2,A3,A4,A5,A6,A7,A8,Th57;
end;
