let s be State of SCMPDS; :: thesis: for P being Instruction-Sequence of SCMPDS
for I, J being Program of
for k being Nat st k <= LifeSpan ((P +* (stop I)),()) & I c= J & I is_closed_on s,P & I is_halting_on s,P holds
IC (Comput ((P +* J),(),k)) in dom (stop I)

let P be Instruction-Sequence of SCMPDS; :: thesis: for I, J being Program of
for k being Nat st k <= LifeSpan ((P +* (stop I)),()) & I c= J & I is_closed_on s,P & I is_halting_on s,P holds
IC (Comput ((P +* J),(),k)) in dom (stop I)

let I, J be Program of ; :: thesis: for k being Nat st k <= LifeSpan ((P +* (stop I)),()) & I c= J & I is_closed_on s,P & I is_halting_on s,P holds
IC (Comput ((P +* J),(),k)) in dom (stop I)

let k be Nat; :: thesis: ( k <= LifeSpan ((P +* (stop I)),()) & I c= J & I is_closed_on s,P & I is_halting_on s,P implies IC (Comput ((P +* J),(),k)) in dom (stop I) )
set ss = Initialize s;
set PP = P +* (stop I);
set s1 = Comput ((P +* J),(),k);
set s2 = Comput ((P +* (stop I)),(),k);
assume that
A1: k <= LifeSpan ((P +* (stop I)),()) and
A2: I c= J and
A3: I is_closed_on s,P and
A4: I is_halting_on s,P ; :: thesis: IC (Comput ((P +* J),(),k)) in dom (stop I)
Comput ((P +* J),(),k) = Comput ((P +* (stop I)),(),k) by A1, A2, A3, A4, Th18;
hence IC (Comput ((P +* J),(),k)) in dom (stop I) by ; :: thesis: verum