let s1, s2 be State of ; :: thesis: for I being InitHalting Program of st Initialized I c= s1 & Initialized I c= s2 & s1,s2 equal_outside NAT holds
( LifeSpan s1 = LifeSpan s2 & Result s1, Result s2 equal_outside NAT )
let I be InitHalting Program of ; :: thesis: ( Initialized I c= s1 & Initialized I c= s2 & s1,s2 equal_outside NAT implies ( LifeSpan s1 = LifeSpan s2 & Result s1, Result s2 equal_outside NAT ) )
assume that
A1: Initialized I c= s1 and
A2: Initialized I c= s2 and
A3: s1,s2 equal_outside NAT ; :: thesis: ( LifeSpan s1 = LifeSpan s2 & Result s1, Result s2 equal_outside NAT )
A4: ProgramPart s2 halts_on s2 by A2, SCM_HALT:5;
A5: ProgramPart s1 halts_on s1 by A1, SCM_HALT:5;

A6: now

let l be Element of NAT ; :: thesis: ( CurInstr (Computation s2,l) = halt SCM+FSA implies LifeSpan s1 <= l )
assume A7: CurInstr (Computation s2,l) = halt SCM+FSA ; :: thesis: LifeSpan s1 <= l
CurInstr (Computation s1,l) = CurInstr (Computation s2,l) by A1, A2, A3, Th11;
hence LifeSpan s1 <= l by A5, A7, AMI_1:def 46; :: thesis: verum

end;

CurInstr (Computation s2,(LifeSpan s1)) = CurInstr (Computation s1,(LifeSpan s1)) by A1, A2, A3, Th11
.= halt SCM+FSA by A5, AMI_1:def 46 ;
hence X: LifeSpan s1 = LifeSpan s2 by A6, A4, AMI_1:def 46; :: thesis: Result s1, Result s2 equal_outside NAT
then ProgramPart s2 halts_on s2 by A2, SCM_HALT:5;
then A8: Result s2 = Computation s2,(LifeSpan s1) by A2, AMI_1:122, SCM_HALT:5, X;
ProgramPart s1 halts_on s1 by A1, SCM_HALT:5;
then Result s1 = Computation s1,(LifeSpan s1) by A1, AMI_1:122, SCM_HALT:5;
hence Result s1, Result s2 equal_outside NAT by A1, A2, A3, A8, Th11; :: thesis: verum