let s be 0 -started State of SCMPDS; :: thesis: for I being parahalting Program of SCMPDS
for J being Program of SCMPDS st stop I c= s holds
for m being Element of NAT st m <= LifeSpan ((ProgramPart s),s) holds
Comput ((ProgramPart s),s,m), Comput ((ProgramPart (s +* (stop (I ';' J)))),(s +* (stop (I ';' J))),m) equal_outside NAT
let I be parahalting Program of SCMPDS; :: thesis: for J being Program of SCMPDS st stop I c= s holds
for m being Element of NAT st m <= LifeSpan ((ProgramPart s),s) holds
Comput ((ProgramPart s),s,m), Comput ((ProgramPart (s +* (stop (I ';' J)))),(s +* (stop (I ';' J))),m) equal_outside NAT
let J be Program of SCMPDS; :: thesis: ( stop I c= s implies for m being Element of NAT st m <= LifeSpan ((ProgramPart s),s) holds
Comput ((ProgramPart s),s,m), Comput ((ProgramPart (s +* (stop (I ';' J)))),(s +* (stop (I ';' J))),m) equal_outside NAT )
assume A1: stop I c= s ; :: thesis: for m being Element of NAT st m <= LifeSpan ((ProgramPart s),s) holds
Comput ((ProgramPart s),s,m), Comput ((ProgramPart (s +* (stop (I ';' J)))),(s +* (stop (I ';' J))),m) equal_outside NAT
set sIJ = stop (I ';' J);
set SS = Stop SCMPDS;
let m be Element of NAT ; :: thesis: ( m <= LifeSpan ((ProgramPart s),s) implies Comput ((ProgramPart s),s,m), Comput ((ProgramPart (s +* (stop (I ';' J)))),(s +* (stop (I ';' J))),m) equal_outside NAT )
assume A2: m <= LifeSpan ((ProgramPart s),s) ; :: thesis: Comput ((ProgramPart s),s,m), Comput ((ProgramPart (s +* (stop (I ';' J)))),(s +* (stop (I ';' J))),m) equal_outside NAT
s +* (stop (I ';' J)) = s +* (I ';' (J ';' (Stop SCMPDS))) by AFINSQ_1:30;
hence Comput ((ProgramPart s),s,m), Comput ((ProgramPart (s +* (stop (I ';' J)))),(s +* (stop (I ';' J))),m) equal_outside NAT by A1, A2, Th23; :: thesis: verum