let P1, P2 be Instruction-Sequence of SCM+FSA; :: thesis: for s1, s2 being 0 -started State of SCM+FSA
for I being Program of SCM+FSA st I is_closed_on s1,P1 & I is_halting_on s1,P1 & I c= P1 & I c= P2 & DataPart s1 = DataPart s2 holds
LifeSpan (P1,s1) = LifeSpan (P2,s2)
let s1, s2 be 0 -started State of SCM+FSA; :: thesis: for I being Program of SCM+FSA st I is_closed_on s1,P1 & I is_halting_on s1,P1 & I c= P1 & I c= P2 & DataPart s1 = DataPart s2 holds
LifeSpan (P1,s1) = LifeSpan (P2,s2)
let J be Program of SCM+FSA; :: thesis: ( J is_closed_on s1,P1 & J is_halting_on s1,P1 & J c= P1 & J c= P2 & DataPart s1 = DataPart s2 implies LifeSpan (P1,s1) = LifeSpan (P2,s2) )
assume that
A1: J is_closed_on s1,P1 and
A2: J is_halting_on s1,P1 and
A3: J c= P1 and
A4: J c= P2 and
A5: DataPart s1 = DataPart s2 ; :: thesis: LifeSpan (P1,s1) = LifeSpan (P2,s2)
A6: P1 = P1 +* J by A3, FUNCT_4:98;
s1 = Initialize s1 by MEMSTR_0:44;
then A7: P1 halts_on s1 by A2, A6, SCMFSA7B:def 7;

A8: now :: thesis: for k being Element of NAT st CurInstr (P2,(Comput (P2,s2,k))) = halt SCM+FSA holds
LifeSpan (P1,s1) <= k

let k be Element of NAT ; :: thesis: ( CurInstr (P2,(Comput (P2,s2,k))) = halt SCM+FSA implies LifeSpan (P1,s1) <= k )
assume CurInstr (P2,(Comput (P2,s2,k))) = halt SCM+FSA ; :: thesis: LifeSpan (P1,s1) <= k
then CurInstr (P1,(Comput (P1,s1,k))) = halt SCM+FSA by A1, A5, Th17, A3, A4;
hence LifeSpan (P1,s1) <= k by A7, EXTPRO_1:def 15; :: thesis: verum

end;

CurInstr (P1,(Comput (P1,s1,(LifeSpan (P1,s1))))) = halt SCM+FSA by A7, EXTPRO_1:def 15;
then A9: CurInstr (P2,(Comput (P2,s2,(LifeSpan (P1,s1))))) = halt SCM+FSA by A1, A5, Th17, A3, A4;
then P2 halts_on s2 by EXTPRO_1:29;
hence LifeSpan (P1,s1) = LifeSpan (P2,s2) by A9, A8, EXTPRO_1:def 15; :: thesis: verum