theorem Th10:
  (p.v)`2 = 7 implies ex w,u being Element of dom p, T,X,Y,y,z st w = v^<*0 *>
  & u = v^<*1*> & (p.v)`1 = [X^T^Y,z] & (p.w)`1 = [T,y] & (p.u)`1 =
  [X^<*y*>^Y,z]
proof
A1: v is correct by Def12;
  assume
A2: (p.v)`2 = 7;
  then
A3: ex T,X,Y,y,z st (p.v)`1 = [X^T^Y,z] & (p.(v^<*0*>))`1 = [T,y] &
  (p.(v^<*1*>))`1 = [X^<*y*>^Y,z] by A1,Def4;
A4: branchdeg v = 2 by A1,A2,Def4;
  then
A5: v^<*0*> in dom p by Th2;
  v^<*1*> in dom p by A4,Th2;
  hence thesis by A3,A5;
end;
