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