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
  f iteration_terminates_for I,s iff iteration-degree(I,s,f) < +infty
proof
  hereby
    assume 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;
    (len r)-1 in REAL by XREAL_0:def 1;
    hence iteration-degree(I,s,f) < +infty by A1,XXREAL_0:9;
  end;
  thus thesis by Def34;
end;
