let A be preIfWhileAlgebra; :: thesis:
let S be non empty set ; :: thesis:
let T be Subset of S; :: thesis:
let s be Element of S; :: thesis:
let f be ExecutionFunction of A,S,T; :: thesis:
let C, I be absolutely-terminating Element of A; :: thesis:
given r being non empty FinSequence of S such that A1: r . 1 = f . (s,C) and
A2: r . (len r) nin T and
A3: for i being Nat st 1 <= i & i < len r holds
( r . i in T & r . (i + 1) = f . ((r . i),(I \; C)) ) ; :: according to AOFA_000:def 33 :: thesis:

let i be Nat; :: thesis:
assume that
A5: 1 <= i and
A6: i < len r ; :: thesis:
thus r . i in T by A3, A5, A6; :: thesis:
hence ( [(r . i),(I \; C)] in TerminatingPrograms (A,S,T,f) & r . (i + 1) = f . ((r . i),(I \; C)) ) by A3, A5, A6, Def36; :: thesis:

end;