let s be State of SCMPDS ; :: thesis: for I being parahalting Program of SCMPDS
for J being Program of SCMPDS st Initialize (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 ((Initialize s) +* (stop (I ';' J)))),((Initialize s) +* (stop (I ';' J))),m equal_outside NAT
let I be parahalting Program of SCMPDS ; :: thesis: for J being Program of SCMPDS st Initialize (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 ((Initialize s) +* (stop (I ';' J)))),((Initialize s) +* (stop (I ';' J))),m equal_outside NAT
let J be Program of SCMPDS ; :: thesis: ( Initialize (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 ((Initialize s) +* (stop (I ';' J)))),((Initialize s) +* (stop (I ';' J))),m equal_outside NAT )
assume A1: Initialize (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 ((Initialize s) +* (stop (I ';' J)))),((Initialize 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 ((Initialize s) +* (stop (I ';' J)))),((Initialize s) +* (stop (I ';' J))),m equal_outside NAT )
assume A2: m <= LifeSpan (ProgramPart s),s ; :: thesis: Comput (ProgramPart s),s,m, Comput (ProgramPart ((Initialize s) +* (stop (I ';' J)))),((Initialize s) +* (stop (I ';' J))),m equal_outside NAT
stop (I ';' J) = I ';' (J ';' (Stop SCMPDS )) by SCMPDS_4:46;
then (Initialize s) +* (stop (I ';' J)) = s +* (I ';' (J ';' (Stop SCMPDS ))) by A1, SCMPDS_4:34;
hence Comput (ProgramPart s),s,m, Comput (ProgramPart ((Initialize s) +* (stop (I ';' J)))),((Initialize s) +* (stop (I ';' J))),m equal_outside NAT by A1, A2, Th23; :: thesis: verum