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 Th83:
  s nin T implies f iteration_terminates_for I,s & iteration-degree(I,s,f) = 0
proof
  assume
A1: s nin T;
  set r = <*s*>;
A2: r.1 = s;
A3: len r = 1 by FINSEQ_1:40;
A4: for i being Nat st 1 <= i & i < len r
  holds r.i in T & r.(i+1) = f.(r.i, I) by FINSEQ_1:40;
A5: 1-1 = 0.;
  thus f iteration_terminates_for I,s
  proof
    take <*s*>;
    thus thesis by A1,A2,FINSEQ_1:40;
  end;
  hence thesis by A1,A2,A3,A4,A5,Def34;
end;
