let s be State of SCMPDS ; :: thesis: for I being parahalting Program of SCMPDS 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 SCMPDS ; :: 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)));
assume A1: 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) )
A2: Initialized (stop I) c= s +* (Initialized (stop I)) by FUNCT_4:26;
A3: I c= s by A1, SCMPDS_4:57;
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)))));
A4: IC (Computation s,(LifeSpan (s +* (Initialized (stop I))))) = IC (Computation (s +* (Initialized (stop I))),(LifeSpan (s +* (Initialized (stop I))))) by A1, Th29, AMI_1:121
.= IC (Computation (s +* (Initialized (stop I))),(LifeSpan ((s +* (Initialized (stop I))) +* (Initialized (stop I))))) by A2, FUNCT_4:79 ;
A5: IC (Computation (s +* (Initialized (stop I))),(LifeSpan ((s +* (Initialized (stop I))) +* (Initialized (stop I))))) in dom (stop I) by A2, SCMPDS_4:def 9;
A6: s +* (Initialized (stop I)) is halting by FUNCT_4:26, SCMPDS_4:63;
stop I c= Initialized (stop I) by SCMPDS_4:9;
then A7: stop I c= s +* (Initialized (stop I)) by A2, XBOOLE_1:1;
reconsider n = IC (Computation (s +* (Initialized (stop I))),(LifeSpan ((s +* (Initialized (stop I))) +* (Initialized (stop I))))) as Element of NAT by ORDINAL1:def 13;
card (stop I) = (card I) + 1 by SCMPDS_4:45, SCMPDS_4:74;
then n < (card I) + 1 by A5, SCMPDS_4:1;
then A9: n <= card I by INT_1:20;

now

per cases ( n < card I or n = card I ) by A9, XXREAL_0:1;

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

then A10: inspos n in dom I by SCMPDS_4:1;
thus halt SCMPDS = CurInstr (Computation (s +* (Initialized (stop I))),(LifeSpan (s +* (Initialized (stop I))))) by A6, 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 A5, A7, GRFUNC_1:8
.= I . (IC (Computation (s +* (Initialized (stop I))),(LifeSpan ((s +* (Initialized (stop I))) +* (Initialized (stop I)))))) by A10, SCMPDS_4:37
.= s . (IC (Computation s,(LifeSpan (s +* (Initialized (stop I)))))) by A3, A4, A10, 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 A4; :: 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