let s be State of SCMPDS ; :: thesis: for I being Program of SCMPDS
for k being Element of NAT st I is_closed_on s & I is_halting_on s & k < LifeSpan (s +* (Initialized (stop I))) holds
IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k) in dom I
let I be Program of SCMPDS ; :: thesis: for k being Element of NAT st I is_closed_on s & I is_halting_on s & k < LifeSpan (s +* (Initialized (stop I))) holds
IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k) in dom I
let k be Element of NAT ; :: thesis: ( I is_closed_on s & I is_halting_on s & k < LifeSpan (s +* (Initialized (stop I))) implies IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k) in dom I )
set IsI = Initialized (stop I);
set ss = s +* (Initialized (stop I));
set m = LifeSpan (s +* (Initialized (stop I)));
set Sp = Stop SCMPDS ;
assume that
A1: I is_closed_on s and
A2: I is_halting_on s and
A3: k < LifeSpan (s +* (Initialized (stop I))) ; :: thesis: IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k) in dom I
set Sk = Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k;
set Ik = IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k);
A4: IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k) in dom (stop I) by A1, Def2;
reconsider n = IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k) as Element of NAT ;
A5: stop I = I ';' (Stop SCMPDS ) by SCMPDS_4:def 7;
( Initialized (stop I) c= s +* (Initialized (stop I)) & stop I c= Initialized (stop I) ) by FUNCT_4:26, SCMPDS_4:9;
then A6: stop I c= s +* (Initialized (stop I)) by XBOOLE_1:1;
A7: ProgramPart (s +* (Initialized (stop I))) halts_on s +* (Initialized (stop I)) by A2, Def3;

A8: now

Y: (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k)) /. (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k)) = (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k) . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k)) by AMI_1:150;
assume A9: n = card I ; :: thesis: contradiction
CurInstr (ProgramPart (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k)),(Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k) = (s +* (Initialized (stop I))) . (IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k)) by AMI_1:54, Y
.= (stop I) . (0 + n) by A4, A6, GRFUNC_1:8
.= halt SCMPDS by A5, A9, SCMPDS_4:38, JJ, KK ;
hence contradiction by A3, A7, AMI_1:def 46; :: thesis: verum

end;

card (stop I) = (card I) + 1 by SCMPDS_5:7;
then n < (card I) + 1 by A4, SCMPDS_4:1;
then n <= card I by INT_1:20;
then n < card I by A8, XXREAL_0:1;
hence IC (Comput (ProgramPart (s +* (Initialized (stop I)))),(s +* (Initialized (stop I))),k) in dom I by SCMPDS_4:1; :: thesis: verum