reserve s for non empty typealg,
  T,X,Y,T9,X9,Y9 for FinSequence of s,
  x,y,z,y9,z9 for type of s;
reserve Tr for PreProof of s;
reserve p for Proof of s,
  v for Element of dom p;

theorem Th2:
  branchdeg v = 2 implies v^<*0*> in dom p & v^<*1*> in dom p
proof
A1: succ v = {v^<*n*> where n is Nat: v^<*n*> in dom p}
  by TREES_2:def 5;
  assume branchdeg v = 2;
  then card succ v = 2 by TREES_2:def 12;
  then consider x,y being object such that
A2: x <> y and
A3: succ v = {x,y} by CARD_2:60;
  x in succ v by A3,TARSKI:def 2;
  then consider n being Nat such that
A4: x = v^<*n*> and
A5: v^<*n*> in dom p by A1;
  y in succ v by A3,TARSKI:def 2;
  then consider k being Nat such that
A6: y = v^<*k*> and
A7: v^<*k*> in dom p by A1;
  n <> 0 or k <> 0 by A2,A4,A6;
  then
A8: n > 0 or k > 0;
  thus v^<*0*> in dom p by A5,TREES_1:def 3;
  n >= 0+1 or k >= 0+1 by A8,NAT_1:13;
  hence thesis by A5,A7,TREES_1:def 3;
end;
