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)),(Initialize s)) & I c= J & I is_closed_on s,P & I is_halting_on s,P holds

IC (Comput ((P +* J),(Initialize s),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)),(Initialize s)) & I c= J & I is_closed_on s,P & I is_halting_on s,P holds

IC (Comput ((P +* J),(Initialize s),k)) in dom (stop I)

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

IC (Comput ((P +* J),(Initialize s),k)) in dom (stop I)

let k be Nat; :: thesis: ( k <= LifeSpan ((P +* (stop I)),(Initialize s)) & I c= J & I is_closed_on s,P & I is_halting_on s,P implies IC (Comput ((P +* J),(Initialize s),k)) in dom (stop I) )

set ss = Initialize s;

set PP = P +* (stop I);

set s1 = Comput ((P +* J),(Initialize s),k);

set s2 = Comput ((P +* (stop I)),(Initialize s),k);

assume that

A1: k <= LifeSpan ((P +* (stop I)),(Initialize s)) 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),(Initialize s),k)) in dom (stop I)

Comput ((P +* J),(Initialize s),k) = Comput ((P +* (stop I)),(Initialize s),k) by A1, A2, A3, A4, Th18;

hence IC (Comput ((P +* J),(Initialize s),k)) in dom (stop I) by A3, SCMPDS_6:def 2; :: thesis: verum

for I, J being Program of

for k being Nat st k <= LifeSpan ((P +* (stop I)),(Initialize s)) & I c= J & I is_closed_on s,P & I is_halting_on s,P holds

IC (Comput ((P +* J),(Initialize s),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)),(Initialize s)) & I c= J & I is_closed_on s,P & I is_halting_on s,P holds

IC (Comput ((P +* J),(Initialize s),k)) in dom (stop I)

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

IC (Comput ((P +* J),(Initialize s),k)) in dom (stop I)

let k be Nat; :: thesis: ( k <= LifeSpan ((P +* (stop I)),(Initialize s)) & I c= J & I is_closed_on s,P & I is_halting_on s,P implies IC (Comput ((P +* J),(Initialize s),k)) in dom (stop I) )

set ss = Initialize s;

set PP = P +* (stop I);

set s1 = Comput ((P +* J),(Initialize s),k);

set s2 = Comput ((P +* (stop I)),(Initialize s),k);

assume that

A1: k <= LifeSpan ((P +* (stop I)),(Initialize s)) 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),(Initialize s),k)) in dom (stop I)

Comput ((P +* J),(Initialize s),k) = Comput ((P +* (stop I)),(Initialize s),k) by A1, A2, A3, A4, Th18;

hence IC (Comput ((P +* J),(Initialize s),k)) in dom (stop I) by A3, SCMPDS_6:def 2; :: thesis: verum