let s be State of SCMPDS ; :: thesis: for I being parahalting Program of SCMPDS
for k being Element of NAT st k < LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) holds
IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k) in dom I
let I be parahalting Program of SCMPDS ; :: thesis: for k being Element of NAT st k < LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) holds
IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k) in dom I
let k be Element of NAT ; :: thesis: ( k < LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) implies IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k) in dom I )
set ss = (Initialize s) +* (stop I);
set m = LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I));
set Sk = Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k;
set Ik = IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k);
I1: s +* (Initialize (stop I)) = (Initialize s) +* (stop I) by SCMPDS_4:5;
A1: ProgramPart ((Initialize s) +* (stop I)) halts_on (Initialize s) +* (stop I) by I1, FUNCT_4:26, SCMPDS_4:63;
reconsider n = IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k) as Element of NAT ;
A2: Initialize (stop I) c= (Initialize s) +* (stop I) by I1, FUNCT_4:26;
then A3: IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k) in dom (stop I) by SCMPDS_4:def 9;
stop I c= Initialize (stop I) by SCMPDS_4:9;
then A4: stop I c= (Initialize s) +* (stop I) by A2, XBOOLE_1:1;
assume A5: k < LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) ; :: thesis: IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k) in dom I

A6: now

Y: (ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)) /. (IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)) = (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k) . (IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)) by COMPOS_1:38;
assume A7: n = card I ; :: thesis: contradiction
y: 0 in dom (Stop SCMPDS ) by COMPOS_1:45;
x: (Stop SCMPDS ) . 0 = halt SCMPDS by AFINSQ_1:38;
TX: ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k) = ProgramPart ((Initialize s) +* (stop I)) by AMI_1:123;
CurInstr (ProgramPart ((Initialize s) +* (stop I))),(Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k) = ((Initialize s) +* (stop I)) . (IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)) by Y, TX, AMI_1:54
.= (stop I) . (0 + n) by A3, A4, GRFUNC_1:8
.= halt SCMPDS by A7, x, y, SCMPDS_4:38 ;
hence contradiction by A5, A1, AMI_1:def 46; :: thesis: verum

end;

card (stop I) = (card I) + 1 by LL, SCMPDS_4:45;
then n < (card I) + 1 by A3, AFINSQ_1:70;
then n <= card I by INT_1:20;
then n < card I by A6, XXREAL_0:1;
hence IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k) in dom I by AFINSQ_1:70; :: thesis: verum