let s1, s2 be State of SCMPDS ; :: thesis: for I being parahalting Program of SCMPDS st Initialize (stop I) c= s1 & Initialize (stop I) c= s2 & s1,s2 equal_outside NAT holds
( LifeSpan (ProgramPart s1),s1 = LifeSpan (ProgramPart s2),s2 & Result (ProgramPart s1),s1, Result (ProgramPart s2),s2 equal_outside NAT )
let I be parahalting Program of SCMPDS ; :: thesis: ( Initialize (stop I) c= s1 & Initialize (stop I) c= s2 & s1,s2 equal_outside NAT implies ( LifeSpan (ProgramPart s1),s1 = LifeSpan (ProgramPart s2),s2 & Result (ProgramPart s1),s1, Result (ProgramPart s2),s2 equal_outside NAT ) )
set SI = stop I;
assume that
A1: Initialize (stop I) c= s1 and
A2: Initialize (stop I) c= s2 and
A3: s1,s2 equal_outside NAT ; :: thesis: ( LifeSpan (ProgramPart s1),s1 = LifeSpan (ProgramPart s2),s2 & Result (ProgramPart s1),s1, Result (ProgramPart s2),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 s2),(Comput (ProgramPart s2),s2,l) = halt SCMPDS implies LifeSpan (ProgramPart s1),s1 <= l )
assume A7: CurInstr (ProgramPart s2),(Comput (ProgramPart s2),s2,l) = halt SCMPDS ; :: thesis: LifeSpan (ProgramPart s1),s1 <= l
CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,l) = CurInstr (ProgramPart s2),(Comput (ProgramPart s2),s2,l) by A1, A2, A3, Th20;
hence LifeSpan (ProgramPart s1),s1 <= l by A5, A7, AMI_1:def 46; :: thesis: verum

end;

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