let P1, P2 be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for s1, s2 being State of SCM+FSA
for I being Program of SCM+FSA st I is_closed_on s1,P1 & I is_halting_on s1,P1 & DataPart s1 = DataPart s2 holds
( LifeSpan ((P1 +* I),(Initialize s1)) = LifeSpan ((P2 +* I),(Initialize s2)) & NPP (Result ((P1 +* I),(Initialize s1))) = NPP (Result ((P2 +* I),(Initialize s2))) )
let s1, s2 be 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 & DataPart s1 = DataPart s2 holds
( LifeSpan ((P1 +* I),(Initialize s1)) = LifeSpan ((P2 +* I),(Initialize s2)) & NPP (Result ((P1 +* I),(Initialize s1))) = NPP (Result ((P2 +* I),(Initialize s2))) )
set D = Data-Locations SCM+FSA;
let I be Program of SCM+FSA; :: thesis: ( I is_closed_on s1,P1 & I is_halting_on s1,P1 & DataPart s1 = DataPart s2 implies ( LifeSpan ((P1 +* I),(Initialize s1)) = LifeSpan ((P2 +* I),(Initialize s2)) & NPP (Result ((P1 +* I),(Initialize s1))) = NPP (Result ((P2 +* I),(Initialize s2))) ) )
assume A1: I is_closed_on s1,P1 ; :: thesis: ( not I is_halting_on s1,P1 or not DataPart s1 = DataPart s2 or ( LifeSpan ((P1 +* I),(Initialize s1)) = LifeSpan ((P2 +* I),(Initialize s2)) & NPP (Result ((P1 +* I),(Initialize s1))) = NPP (Result ((P2 +* I),(Initialize s2))) ) )
set ss2 = Initialize s2;
set PP2 = P2 +* I;
set ss1 = Initialize s1;
set PP1 = P1 +* I;
assume A3: I is_halting_on s1,P1 ; :: thesis: ( not DataPart s1 = DataPart s2 or ( LifeSpan ((P1 +* I),(Initialize s1)) = LifeSpan ((P2 +* I),(Initialize s2)) & NPP (Result ((P1 +* I),(Initialize s1))) = NPP (Result ((P2 +* I),(Initialize s2))) ) )
then A4: P1 +* I halts_on Initialize s1 by SCMFSA7B:def 8;
then A5: Result ((P1 +* I),(Initialize s1)) = Comput ((P1 +* I),(Initialize s1),(LifeSpan ((P1 +* I),(Initialize s1)))) by EXTPRO_1:23;
assume A6: DataPart s1 = DataPart s2 ; :: thesis: ( LifeSpan ((P1 +* I),(Initialize s1)) = LifeSpan ((P2 +* I),(Initialize s2)) & NPP (Result ((P1 +* I),(Initialize s1))) = NPP (Result ((P2 +* I),(Initialize s2))) )
then I is_halting_on s2,P2 by A1, A3, SCMFSA8B:8;
then A7: P2 +* I halts_on Initialize s2 by SCMFSA7B:def 8;

A8: now

let l be Element of NAT ; :: thesis: ( CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),l))) = halt SCM+FSA implies LifeSpan ((P1 +* I),(Initialize s1)) <= l )
assume A9: CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),l))) = halt SCM+FSA ; :: thesis: LifeSpan ((P1 +* I),(Initialize s1)) <= l
CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),l))) = CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),l))) by A1, A3, A6, Th100;
hence LifeSpan ((P1 +* I),(Initialize s1)) <= l by A4, A9, EXTPRO_1:def 14; :: thesis: verum

end;

CurInstr ((P2 +* I),(Comput ((P2 +* I),(Initialize s2),(LifeSpan ((P1 +* I),(Initialize s1)))))) = CurInstr ((P1 +* I),(Comput ((P1 +* I),(Initialize s1),(LifeSpan ((P1 +* I),(Initialize s1)))))) by A1, A3, A6, Th100
.= halt SCM+FSA by A4, EXTPRO_1:def 14 ;
hence LifeSpan ((P1 +* I),(Initialize s1)) = LifeSpan ((P2 +* I),(Initialize s2)) by A8, A7, EXTPRO_1:def 14; :: thesis: NPP (Result ((P1 +* I),(Initialize s1))) = NPP (Result ((P2 +* I),(Initialize s2)))
then Result ((P2 +* I),(Initialize s2)) = Comput ((P2 +* I),(Initialize s2),(LifeSpan ((P1 +* I),(Initialize s1)))) by A7, EXTPRO_1:23;
hence NPP (Result ((P1 +* I),(Initialize s1))) = NPP (Result ((P2 +* I),(Initialize s2))) by A1, A3, A6, A5, Th100; :: thesis: verum