let s1, s2 be State of SCMPDS ; :: thesis: for I being Program of SCMPDS st I is_closed_on s1 & I is_halting_on s1 & Initialize (stop I) c= s1 & Initialize (stop I) c= s2 & DataPart s1 = DataPart s2 holds
LifeSpan (ProgramPart s1),s1 = LifeSpan (ProgramPart s2),s2
let I be Program of SCMPDS ; :: thesis: ( I is_closed_on s1 & I is_halting_on s1 & Initialize (stop I) c= s1 & Initialize (stop I) c= s2 & DataPart s1 = DataPart s2 implies LifeSpan (ProgramPart s1),s1 = LifeSpan (ProgramPart s2),s2 )
assume that
A1: I is_closed_on s1 and
A2: I is_halting_on s1 and
A3: Initialize (stop I) c= s1 and
A4: Initialize (stop I) c= s2 and
A5: DataPart s1 = DataPart s2 ; :: thesis: LifeSpan (ProgramPart s1),s1 = LifeSpan (ProgramPart s2),s2
I1: s1 +* (Initialize (stop I)) = (Initialize s1) +* (stop I) by SCMPDS_4:5;
s1 = (Initialize s1) +* (stop I) by A3, I1, FUNCT_4:79;
then A6: ProgramPart s1 halts_on s1 by A2, SCMPDS_6:def 3;

A7: now

let k be Element of NAT ; :: thesis: ( CurInstr (ProgramPart s2),(Comput (ProgramPart s2),s2,k) = halt SCMPDS implies LifeSpan (ProgramPart s1),s1 <= k )
assume CurInstr (ProgramPart s2),(Comput (ProgramPart s2),s2,k) = halt SCMPDS ; :: thesis: LifeSpan (ProgramPart s1),s1 <= k
then CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,k) = halt SCMPDS by A1, A3, A4, A5, Th24;
hence LifeSpan (ProgramPart s1),s1 <= k by A6, AMI_1:def 46; :: thesis: verum

end;

CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1)) = halt SCMPDS by A6, AMI_1:def 46;
then A8: CurInstr (ProgramPart s2),(Comput (ProgramPart s2),s2,(LifeSpan (ProgramPart s1),s1)) = halt SCMPDS by A1, A3, A4, A5, Th24;
then ProgramPart s2 halts_on s2 by AMI_1:146;
hence LifeSpan (ProgramPart s1),s1 = LifeSpan (ProgramPart s2),s2 by A8, A7, AMI_1:def 46; :: thesis: verum