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 Th68:
  for T1,T2 being Tree, x being object holds x in tree(T1,T2) iff x = {} or
  ex p st p in T1 & x = <*0*>^p or p in T2 & x = <*1*>^p
proof
  let T1, T2 be Tree;
  let x;
  set p = <*T1,T2*>;
A1: len p = 2 by FINSEQ_1:44;
  thus x in tree(T1,T2) & x <> {} implies
  ex p st p in T1 & x = <*0*>^p or p in T2 & x = <*1*>^p
  proof
    assume that
A4: x in tree(T1,T2) and
A5: x <> {};
    consider n, q such that
A6: n < len p and
A7: q in p.(n+1) and
A8: x = <*n*>^q by A4,A5,Def15;
    1+1 = 2;
    then n <= 1 by A1,A6,NAT_1:13;
    then n = 1 or n < 0+1 by XXREAL_0:1;
    then n = 1 or n = 0 by NAT_1:13;
    hence thesis by A7,A8;
  end;
  now
    given q such that
A9: q in T1 & x = <*0*>^q or q in T2 & x = <*1*>^q;
    x = <*0*>^q or x <> <*0*>^q;
    then consider n such that
A10: n = 0 & x = <*0*>^q or n = 1 & x <> <*0*>^q;
    take n,q;
    thus n < len p by A1,A10;
    (<*0*>^q).1 = 0 by FINSEQ_1:41;
    hence q in p.(n+1) & x = <*n*>^q by A9,A10,FINSEQ_1:41;
  end;
  hence thesis by Def15;
end;
