let P1, P2 be Instruction-Sequence of SCMPDS; :: thesis: for s being 0 -started State of SCMPDS
for I being Program of SCMPDS st I is_closed_on s,P1 & I is_halting_on s,P1 & stop I c= P1 & stop I c= P2 holds
( LifeSpan (P1,s) = LifeSpan (P2,s) & Result (P1,s) = Result (P2,s) )
let s be 0 -started State of SCMPDS; :: thesis: for I being Program of SCMPDS st I is_closed_on s,P1 & I is_halting_on s,P1 & stop I c= P1 & stop I c= P2 holds
( LifeSpan (P1,s) = LifeSpan (P2,s) & Result (P1,s) = Result (P2,s) )
let I be Program of SCMPDS; :: thesis: ( I is_closed_on s,P1 & I is_halting_on s,P1 & stop I c= P1 & stop I c= P2 implies ( LifeSpan (P1,s) = LifeSpan (P2,s) & Result (P1,s) = Result (P2,s) ) )
assume that
A1: I is_closed_on s,P1 and
A2: I is_halting_on s,P1 and
A3: stop I c= P1 and
A4: stop I c= P2 ; :: thesis: ( LifeSpan (P1,s) = LifeSpan (P2,s) & Result (P1,s) = Result (P2,s) )
B3: Start-At (0,SCMPDS) c= s by MEMSTR_0:29;
B4: Start-At (0,SCMPDS) c= s by MEMSTR_0:29;
DataPart s = DataPart s ;
then A6: I is_halting_on s,P2 by A1, A2, SCMPDS_6:23;
A7: Initialize s = s by B3, FUNCT_4:98;
A8: Initialize s = s by B4, FUNCT_4:98;
P2 = P2 +* (stop I) by A4, FUNCT_4:98;
then A9: P2 halts_on s by A6, A8, SCMPDS_6:def 3;
P1 = P1 +* (stop I) by A3, FUNCT_4:98;
then A10: P1 halts_on s by A2, A7, SCMPDS_6:def 3;

A11: now

let l be Element of NAT ; :: thesis: ( CurInstr (P2,(Comput (P2,s,l))) = halt SCMPDS implies LifeSpan (P1,s) <= l )
assume A12: CurInstr (P2,(Comput (P2,s,l))) = halt SCMPDS ; :: thesis: LifeSpan (P1,s) <= l
CurInstr (P1,(Comput (P1,s,l))) = CurInstr (P2,(Comput (P2,s,l))) by A1, A3, A4, Th26;
hence LifeSpan (P1,s) <= l by A10, A12, EXTPRO_1:def 15; :: thesis: verum

end;

CurInstr (P2,(Comput (P2,s,(LifeSpan (P1,s))))) = CurInstr (P1,(Comput (P1,s,(LifeSpan (P1,s))))) by A1, A3, A4, Th26
.= halt SCMPDS by A10, EXTPRO_1:def 15 ;
hence LifeSpan (P1,s) = LifeSpan (P2,s) by A11, A9, EXTPRO_1:def 15; :: thesis: Result (P1,s) = Result (P2,s)
then A13: Result (P2,s) = Comput (P2,s,(LifeSpan (P1,s))) by A9, EXTPRO_1:23;
Result (P1,s) = Comput (P1,s,(LifeSpan (P1,s))) by A10, EXTPRO_1:23;
hence Result (P1,s) = Result (P2,s) by A1, A3, A4, A13, Th26; :: thesis: verum