let s be State of ; :: thesis: for I, J being Program of st I c= J & I is_closed_on s & I is_halting_on s & not CurInstr (Computation (s +* (Initialized J)),(LifeSpan (s +* (Initialized (stop I))))) = halt SCMPDS holds
IC (Computation (s +* (Initialized J)),(LifeSpan (s +* (Initialized (stop I))))) = inspos (card I)
let I, J be Program of ; :: thesis: ( I c= J & I is_closed_on s & I is_halting_on s & not CurInstr (Computation (s +* (Initialized J)),(LifeSpan (s +* (Initialized (stop I))))) = halt SCMPDS implies IC (Computation (s +* (Initialized J)),(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 s0 = s +* (Initialized J);
set s1 = Computation (s +* (Initialized J)),(LifeSpan (s +* (Initialized (stop I))));
set s2 = Computation (s +* (Initialized (stop I))),(LifeSpan (s +* (Initialized (stop I))));
set Ik = IC (Computation (s +* (Initialized (stop I))),(LifeSpan (s +* (Initialized (stop I)))));
assume that
A1: I c= J and
A2: I is_closed_on s and
A3: I is_halting_on s ; :: thesis: ( CurInstr (Computation (s +* (Initialized J)),(LifeSpan (s +* (Initialized (stop I))))) = halt SCMPDS or IC (Computation (s +* (Initialized J)),(LifeSpan (s +* (Initialized (stop I))))) = inspos (card I) )
A4: dom I c= dom J by A1, GRFUNC_1:8;
reconsider n = IC (Computation (s +* (Initialized (stop I))),(LifeSpan (s +* (Initialized (stop I))))) as Element of NAT by ORDINAL1:def 13;
A5: stop I c= Initialized (stop I) by SCMPDS_4:9;
Initialized (stop I) c= s +* (Initialized (stop I)) by FUNCT_4:26;
then A6: stop I c= s +* (Initialized (stop I)) by A5, XBOOLE_1:1;
A7: ProgramPart (s +* (Initialized (stop I))) halts_on s +* (Initialized (stop I)) by A3, SCMPDS_6:def 3;
A8: IC (Computation (s +* (Initialized (stop I))),(LifeSpan (s +* (Initialized (stop I))))) in dom (stop I) by A2, SCMPDS_6:def 2;
card (stop I) = (card I) + 1 by SCMPDS_5:7;
then n < (card I) + 1 by A8, SCMPDS_4:1;
then A9: n <= card I by INT_1:20;
A10: IC (Computation (s +* (Initialized J)),(LifeSpan (s +* (Initialized (stop I))))) = IC (Computation (s +* (Initialized (stop I))),(LifeSpan (s +* (Initialized (stop I))))) by A1, A2, A3, Th39, AMI_1:121;

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 +* (Initialized J)),(LifeSpan (s +* (Initialized (stop I)))))

then A11: inspos n in dom I by SCMPDS_4:1;
thus halt SCMPDS = CurInstr (Computation (s +* (Initialized (stop I))),(LifeSpan (s +* (Initialized (stop I))))) by A7, AMI_1:def 46
.= (s +* (Initialized (stop I))) . (IC (Computation (s +* (Initialized (stop I))),(LifeSpan (s +* (Initialized (stop I)))))) by AMI_1:54
.= (stop I) . (IC (Computation (s +* (Initialized (stop I))),(LifeSpan (s +* (Initialized (stop I)))))) by A8, A6, GRFUNC_1:8
.= I . (IC (Computation (s +* (Initialized (stop I))),(LifeSpan (s +* (Initialized (stop I)))))) by A11, SCMPDS_4:37
.= J . (IC (Computation (s +* (Initialized (stop I))),(LifeSpan (s +* (Initialized (stop I)))))) by A1, A11, GRFUNC_1:8
.= (s +* (Initialized J)) . (IC (Computation (s +* (Initialized J)),(LifeSpan (s +* (Initialized (stop I)))))) by A4, A10, A11, Th10
.= CurInstr (Computation (s +* (Initialized J)),(LifeSpan (s +* (Initialized (stop I))))) by AMI_1:54 ; :: thesis: verum

end;

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

hence IC (Computation (s +* (Initialized J)),(LifeSpan (s +* (Initialized (stop I))))) = inspos (card I) by A1, A2, A3, Th39, AMI_1:121; :: thesis: verum

end;

end;

end;

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