let s be State of SCMPDS; :: thesis: for I, J being Program of SCMPDS
for k being Element of NAT st k <= LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I))) & I c= J & I is_closed_on s & I is_halting_on s holds
IC (Comput ((ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),k)) in dom (stop I)
let I, J be Program of SCMPDS; :: thesis: for k being Element of NAT st k <= LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I))) & I c= J & I is_closed_on s & I is_halting_on s holds
IC (Comput ((ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),k)) in dom (stop I)
let k be Element of NAT ; :: thesis: ( k <= LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I))) & I c= J & I is_closed_on s & I is_halting_on s implies IC (Comput ((ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),k)) in dom (stop I) )
set ss = (Initialize s) +* (stop I);
set s1 = Comput ((ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),k);
set s2 = Comput ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k);
assume that
A1: k <= LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I))) and
A2: I c= J and
A3: I is_closed_on s and
A4: I is_halting_on s ; :: thesis: IC (Comput ((ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),k)) in dom (stop I)
IC (Comput ((ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),k)) = IC (Comput ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)) by A1, A2, A3, A4, Th39, COMPOS_1:24;
hence IC (Comput ((ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),k)) in dom (stop I) by A3, SCMPDS_6:def 2; :: thesis: verum