let P1, P2 be the Instructions of SCMPDS -valued ManySortedSet of NAT ; :: thesis: for s1, s2 being State of SCMPDS
for I being Program of SCMPDS st I is_closed_on s1,P1 & I is_halting_on s1,P1 & stop I c= P1 & stop I c= P2 & Start-At (0,SCMPDS) c= s1 & Start-At (0,SCMPDS) c= s2 & DataPart s1 = DataPart s2 holds
LifeSpan (P1,s1) = LifeSpan (P2,s2)
let s1, s2 be State of SCMPDS; :: thesis: for I being Program of SCMPDS st I is_closed_on s1,P1 & I is_halting_on s1,P1 & stop I c= P1 & stop I c= P2 & Start-At (0,SCMPDS) c= s1 & Start-At (0,SCMPDS) c= s2 & DataPart s1 = DataPart s2 holds
LifeSpan (P1,s1) = LifeSpan (P2,s2)
let I be Program of SCMPDS; :: thesis: ( I is_closed_on s1,P1 & I is_halting_on s1,P1 & stop I c= P1 & stop I c= P2 & Start-At (0,SCMPDS) c= s1 & Start-At (0,SCMPDS) c= s2 & DataPart s1 = DataPart s2 implies LifeSpan (P1,s1) = LifeSpan (P2,s2) )
assume that
A1: I is_closed_on s1,P1 and
A2: I is_halting_on s1,P1 and
A3: stop I c= P1 and
A4: stop I c= P2 and
B3: Start-At (0,SCMPDS) c= s1 and
B4: Start-At (0,SCMPDS) c= s2 and
A5: DataPart s1 = DataPart s2 ; :: thesis: LifeSpan (P1,s1) = LifeSpan (P2,s2)
A6: s1 = Initialize s1 by B3, FUNCT_4:104;
P1 = P1 +* (stop I) by A3, FUNCT_4:104;
then A7: P1 halts_on s1 by A2, SCMPDS_6:def 3, A6;

A8: now

let k be Element of NAT ; :: thesis: ( CurInstr (P2,(Comput (P2,s2,k))) = halt SCMPDS implies LifeSpan (P1,s1) <= k )
assume CurInstr (P2,(Comput (P2,s2,k))) = halt SCMPDS ; :: thesis: LifeSpan (P1,s1) <= k
then CurInstr (P1,(Comput (P1,s1,k))) = halt SCMPDS by A1, A3, A4, A5, Th24, B3, B4;
hence LifeSpan (P1,s1) <= k by A7, EXTPRO_1:def 14; :: thesis: verum

end;

CurInstr (P1,(Comput (P1,s1,(LifeSpan (P1,s1))))) = halt SCMPDS by A7, EXTPRO_1:def 14;
then A9: CurInstr (P2,(Comput (P2,s2,(LifeSpan (P1,s1))))) = halt SCMPDS by A1, A3, A4, A5, Th24, B3, B4;
then P2 halts_on s2 by EXTPRO_1:30;
hence LifeSpan (P1,s1) = LifeSpan (P2,s2) by A9, A8, EXTPRO_1:def 14; :: thesis: verum