reserve A for preIfWhileAlgebra,
  C,I,J for Element of A;
reserve S for non empty set,
  T for Subset of S,
  s for Element of S;
reserve f for ExecutionFunction of A,S,T;

theorem
  iteration-degree(I,s,f) >= 0 proof per cases;
  suppose not f iteration_terminates_for I,s;
    hence thesis by Def34;
  end;
  suppose f iteration_terminates_for I,s;
    then consider r being non empty FinSequence of S such that
A1: iteration-degree(I,s,f) = (len r)-1 and r.1 = s
    and r.len r nin T
    and for i being Nat st 1 <= i & i < len r holds r.i in T & r.(i+1) = f.(
    r.i, I)
    by Def34;
    ex i being Nat st len r = i+1 by NAT_1:6;
    hence thesis by A1;
  end;
end;
