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;
