let s1, s2 be State of SCMPDS ; :: thesis: for I being parahalting Program of SCMPDS st Initialized (stop I) c= s1 & Initialized (stop I) c= s2 & s1,s2 equal_outside NAT holds
( LifeSpan s1 = LifeSpan s2 & Result s1, Result s2 equal_outside NAT )
let I be parahalting Program of SCMPDS ; :: thesis: ( Initialized (stop I) c= s1 & Initialized (stop I) c= s2 & s1,s2 equal_outside NAT implies ( LifeSpan s1 = LifeSpan s2 & Result s1, Result s2 equal_outside NAT ) )
set SI = stop I;
assume that
A1: Initialized (stop I) c= s1 and
A2: Initialized (stop 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, SCMPDS_4:63;
A5: ProgramPart s1 halts_on s1 by A1, SCMPDS_4:63;

A6: now

let l be Element of NAT ; :: thesis: ( CurInstr (ProgramPart (Comput (ProgramPart s2),s2,l)),(Comput (ProgramPart s2),s2,l) = halt SCMPDS implies LifeSpan s1 <= l )
assume A7: CurInstr (ProgramPart (Comput (ProgramPart s2),s2,l)),(Comput (ProgramPart s2),s2,l) = halt SCMPDS ; :: thesis: LifeSpan s1 <= l
CurInstr (ProgramPart (Comput (ProgramPart s1),s1,l)),(Comput (ProgramPart s1),s1,l) = CurInstr (ProgramPart (Comput (ProgramPart s2),s2,l)),(Comput (ProgramPart s2),s2,l) by A1, A2, A3, Th20;
hence LifeSpan s1 <= l by A5, A7, AMI_1:def 46; :: thesis: verum

end;

CurInstr (ProgramPart (Comput (ProgramPart s2),s2,(LifeSpan s1))),(Comput (ProgramPart s2),s2,(LifeSpan s1)) = CurInstr (ProgramPart (Comput (ProgramPart s1),s1,(LifeSpan s1))),(Comput (ProgramPart s1),s1,(LifeSpan s1)) by A1, A2, A3, Th20
.= halt SCMPDS by A5, AMI_1:def 46 ;
hence LifeSpan s1 = LifeSpan s2 by A6, A4, AMI_1:def 46; :: thesis: Result s1, Result s2 equal_outside NAT
then A8: Result s2 = Comput (ProgramPart s2),s2,(LifeSpan s1) by A4, AMI_1:122;
Result s1 = Comput (ProgramPart s1),s1,(LifeSpan s1) by A5, AMI_1:122;
hence Result s1, Result s2 equal_outside NAT by A1, A2, A3, A8, Th20; :: thesis: verum