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
  (p.v)`2 = 5 implies ex w being Element of dom p, X,x,y,Y st w = v^<*0*> &
  (p.v)`1 = [X^<*x*y*>^Y,z] & (p.w)`1 = [X^<*x*>^<*y*>^Y,z]
proof
A1: v is correct by Def12;
  assume
A2: (p.v)`2 = 5;
  then
A3: ex X,x,y,Y st (p.v)`1 = [X^<*x*y*>^Y,z] & (p.(v^<*0*>))`1 =
  [X^<*x*>^<*y*>^Y,z] by A1,Def4;
  branchdeg v = 1 by A1,A2,Def4;
  then v^<*0*> in dom p by Th1;
  hence thesis by A3;
end;
