let s be State of ; :: thesis: for I being parahalting Program of holds
( not Initialized I c= s or CurInstr (Computation s,(LifeSpan (s +* (Initialized (stop I))))) = halt SCMPDS or IC (Computation s,(LifeSpan (s +* (Initialized (stop I))))) = inspos (card I) )
let I be parahalting Program of ; :: thesis: ( not Initialized I c= s or CurInstr (Computation s,(LifeSpan (s +* (Initialized (stop I))))) = halt SCMPDS or IC (Computation s,(LifeSpan (s +* (Initialized (stop I))))) = inspos (card I) )
set IsI = Initialized (stop I);
set ss = s +* (Initialized (stop I));
set m = LifeSpan (s +* (Initialized (stop I)));
set s1 = Computation s,(LifeSpan (s +* (Initialized (stop I))));
set s2 = Computation (s +* (Initialized (stop I))),(LifeSpan ((s +* (Initialized (stop I))) +* (Initialized (stop I))));
set Ik = IC (Computation (s +* (Initialized (stop I))),(LifeSpan ((s +* (Initialized (stop I))) +* (Initialized (stop I)))));
A1: ProgramPart (s +* (Initialized (stop I))) halts_on s +* (Initialized (stop I)) by FUNCT_4:26, SCMPDS_4:63;
reconsider n = IC (Computation (s +* (Initialized (stop I))),(LifeSpan ((s +* (Initialized (stop I))) +* (Initialized (stop I))))) as Element of NAT by ORDINAL1:def 13;
A2: Initialized (stop I) c= s +* (Initialized (stop I)) by FUNCT_4:26;
then A3: IC (Computation (s +* (Initialized (stop I))),(LifeSpan ((s +* (Initialized (stop I))) +* (Initialized (stop I))))) in dom (stop I) by SCMPDS_4:def 9;
card (stop I) = (card I) + 1 by SCMPDS_4:45, SCMPDS_4:74;
then n < (card I) + 1 by A3, SCMPDS_4:1;
then A4: n <= card I by INT_1:20;
stop I c= Initialized (stop I) by SCMPDS_4:9;
then A5: stop I c= s +* (Initialized (stop I)) by A2, XBOOLE_1:1;
assume A6: Initialized I c= s ; :: thesis: ( CurInstr (Computation s,(LifeSpan (s +* (Initialized (stop I))))) = halt SCMPDS or IC (Computation s,(LifeSpan (s +* (Initialized (stop I))))) = inspos (card I) )
then A7: IC (Computation s,(LifeSpan (s +* (Initialized (stop I))))) = IC (Computation (s +* (Initialized (stop I))),(LifeSpan (s +* (Initialized (stop I))))) by Th29, AMI_1:121
.= IC (Computation (s +* (Initialized (stop I))),(LifeSpan ((s +* (Initialized (stop I))) +* (Initialized (stop I))))) by A2, FUNCT_4:79 ;
A8: I c= s by A6, SCMPDS_4:57;

now

per cases ( n < card I or n = card I ) by A4, XXREAL_0:1;

case n < card I ; :: thesis: halt SCMPDS = CurInstr (Computation s,(LifeSpan (s +* (Initialized (stop I)))))

then A9: inspos n in dom I by SCMPDS_4:1;
thus halt SCMPDS = CurInstr (Computation (s +* (Initialized (stop I))),(LifeSpan (s +* (Initialized (stop I))))) by A1, AMI_1:def 46
.= CurInstr (Computation (s +* (Initialized (stop I))),(LifeSpan ((s +* (Initialized (stop I))) +* (Initialized (stop I))))) by A2, FUNCT_4:79
.= (s +* (Initialized (stop I))) . (IC (Computation (s +* (Initialized (stop I))),(LifeSpan ((s +* (Initialized (stop I))) +* (Initialized (stop I)))))) by AMI_1:54
.= (stop I) . (IC (Computation (s +* (Initialized (stop I))),(LifeSpan ((s +* (Initialized (stop I))) +* (Initialized (stop I)))))) by A3, A5, GRFUNC_1:8
.= I . (IC (Computation (s +* (Initialized (stop I))),(LifeSpan ((s +* (Initialized (stop I))) +* (Initialized (stop I)))))) by A9, SCMPDS_4:37
.= s . (IC (Computation s,(LifeSpan (s +* (Initialized (stop I)))))) by A8, A7, A9, GRFUNC_1:8
.= CurInstr (Computation s,(LifeSpan (s +* (Initialized (stop I))))) by AMI_1:54 ; :: thesis: verum

end;

case n = card I ; :: thesis: IC (Computation s,(LifeSpan (s +* (Initialized (stop I))))) = inspos (card I)

hence IC (Computation s,(LifeSpan (s +* (Initialized (stop I))))) = inspos (card I) by A7; :: thesis: verum

end;

end;

end;

hence ( CurInstr (Computation s,(LifeSpan (s +* (Initialized (stop I))))) = halt SCMPDS or IC (Computation s,(LifeSpan (s +* (Initialized (stop I))))) = inspos (card I) ) ; :: thesis: verum