let s1, s2 be State of SCMPDS ; :: thesis: for I being Program of SCMPDS st I is_closed_on s1 & I is_halting_on s1 & 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 Program of SCMPDS ; :: thesis: ( I is_closed_on s1 & I is_halting_on s1 & 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 ) )
assume that
A1: I is_closed_on s1 and
A2: I is_halting_on s1 and
A3: Initialize (stop I) c= s1 and
A4: Initialize (stop I) c= s2 and
A5: 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 )
DataPart s1 = DataPart s2 by A5, SCMPDS_6:4;
then A6: I is_halting_on s2 by A1, A2, SCMPDS_6:37;
I1: (Initialize s1) +* (stop I) = s1 +* (Initialize (stop I)) by SCMPDS_4:5;
I2: (Initialize s2) +* (stop I) = s2 +* (Initialize (stop I)) by SCMPDS_4:5;
s2 = (Initialize s2) +* (stop I) by A4, I2, FUNCT_4:79;
then A7: ProgramPart s2 halts_on s2 by A6, SCMPDS_6:def 3;
s1 = (Initialize s1) +* (stop I) by A3, I1, FUNCT_4:79;
then A8: ProgramPart s1 halts_on s1 by A2, SCMPDS_6:def 3;

A9: 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 A10: 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, A3, A4, A5, Th26;
hence LifeSpan (ProgramPart s1),s1 <= l by A8, A10, 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, A3, A4, A5, Th26
.= halt SCMPDS by A8, AMI_1:def 46 ;
hence LifeSpan (ProgramPart s1),s1 = LifeSpan (ProgramPart s2),s2 by A9, A7, AMI_1:def 46; :: thesis: Result (ProgramPart s1),s1, Result (ProgramPart s2),s2 equal_outside NAT
then A11: Result (ProgramPart s2),s2 = Comput (ProgramPart s2),s2,(LifeSpan (ProgramPart s1),s1) by A7, AMI_1:122;
Result (ProgramPart s1),s1 = Comput (ProgramPart s1),s1,(LifeSpan (ProgramPart s1),s1) by A8, AMI_1:122;
hence Result (ProgramPart s1),s1, Result (ProgramPart s2),s2 equal_outside NAT by A1, A3, A4, A5, A11, Th26; :: thesis: verum