let s1, s2 be State of SCM+FSA; :: thesis: for I being Program of SCM+FSA st I is_closed_on s1 & I is_halting_on s1 & I +* (Start-At (0,SCM+FSA)) c= s1 & I +* (Start-At (0,SCM+FSA)) c= s2 & DataPart s1 = DataPart s2 holds
LifeSpan ((ProgramPart s1),s1) = LifeSpan ((ProgramPart s2),s2)
let J be Program of SCM+FSA; :: thesis: ( J is_closed_on s1 & J is_halting_on s1 & J +* (Start-At (0,SCM+FSA)) c= s1 & J +* (Start-At (0,SCM+FSA)) c= s2 & DataPart s1 = DataPart s2 implies LifeSpan ((ProgramPart s1),s1) = LifeSpan ((ProgramPart s2),s2) )
assume that
A1: J is_closed_on s1 and
A2: J is_halting_on s1 and
A3: J +* (Start-At (0,SCM+FSA)) c= s1 and
A4: J +* (Start-At (0,SCM+FSA)) c= s2 and
A5: DataPart s1 = DataPart s2 ; :: thesis: LifeSpan ((ProgramPart s1),s1) = LifeSpan ((ProgramPart s2),s2)
s1 = s1 +* (J +* (Start-At (0,SCM+FSA))) by A3, FUNCT_4:79;
then A6: ProgramPart s1 halts_on s1 by A2, SCMFSA7B:def 8;

A7: now

let k be Element of NAT ; :: thesis: ( CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,k))) = halt SCM+FSA implies LifeSpan ((ProgramPart s1),s1) <= k )
TX2: ProgramPart s2 = ProgramPart (Comput ((ProgramPart s2),s2,k)) by AMI_1:123;
TX1: ProgramPart s1 = ProgramPart (Comput ((ProgramPart s1),s1,k)) by AMI_1:123;
assume CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,k))) = halt SCM+FSA ; :: thesis: LifeSpan ((ProgramPart s1),s1) <= k
then CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,k))) = halt SCM+FSA by A1, A3, A4, A5, Th43, TX1, TX2;
hence LifeSpan ((ProgramPart s1),s1) <= k by A6, EXTPRO_1:def 14; :: thesis: verum

end;

TX1: ProgramPart s1 = ProgramPart (Comput ((ProgramPart s1),s1,(LifeSpan ((ProgramPart s1),s1)))) by AMI_1:123;
TX2: ProgramPart s2 = ProgramPart (Comput ((ProgramPart s2),s2,(LifeSpan ((ProgramPart s1),s1)))) by AMI_1:123;
CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,(LifeSpan ((ProgramPart s1),s1))))) = halt SCM+FSA by A6, EXTPRO_1:def 14;
then A8: CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,(LifeSpan ((ProgramPart s1),s1))))) = halt SCM+FSA by A1, A3, A4, A5, Th43, TX1, TX2;
then ProgramPart s2 halts_on s2 by EXTPRO_1:30;
hence LifeSpan ((ProgramPart s1),s1) = LifeSpan ((ProgramPart s2),s2) by A8, A7, EXTPRO_1:def 14; :: thesis: verum