let P be Instruction-Sequence of SCMPDS; :: thesis:
let s be State of SCMPDS; :: thesis:
let I be shiftable Program of SCMPDS; :: thesis:
let a be Int_position ; :: thesis:
let k1 be Integer; :: thesis:
set b = DataLoc ((s . a),k1);
assume A1: s . (DataLoc ((s . a),k1)) = 0 ; :: thesis:
set i = (a,k1) <>0_goto ((card I) + 1);
set IF = if=0 (a,k1,I);
set pIF = stop (if=0 (a,k1,I));
set pI = stop I;
set s2 = Initialize s;
set s3 = Initialize s;
set P2 = P +* (stop I);
set P3 = P +* (stop (if=0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1);
set P4 = P +* (stop (if=0 (a,k1,I)));
A4: IC (Initialize s) = 0 by MEMSTR_0:47;
A5: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
A6: Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if=0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) <>0_goto ((card I) + 1)),(Initialize s)) by Th22 ;
for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)) . a by A6, SCMPDS_2:55;
then A8: DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)) by SCMPDS_4:8;
not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= 0 by A1, A5, FUNCT_4:11 ;
then A9: IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)) = succ (IC (Initialize s)) by A6, SCMPDS_2:55
.= 0 + 1 by A4 ;
assume A10: I is_closed_on s,P ; :: thesis:
then A11: I is_closed_on Initialize s,P +* (stop I) by Th38;
assume I is_halting_on s,P ; :: thesis:
then A12: P +* (stop I) halts_on Initialize s by Def3;
A13: 0 in dom (stop (if=0 (a,k1,I))) by COMPOS_1:36;
A14: ( Start-At (0,SCMPDS) c= Initialize s & Shift ((stop I),1) c= P +* (stop (if=0 (a,k1,I))) ) by Lm6, FUNCT_4:25;
UU: stop I c= P +* (stop I) by FUNCT_4:25;
A15: card (stop (if=0 (a,k1,I))) = (card (if=0 (a,k1,I))) + 1 by COMPOS_1:55
.= ((card I) + 1) + 1 by Th15 ;

now

let k be Element of NAT ; :: thesis:

per cases ;

suppose 0 < k ; :: thesis:

then consider k1 being Nat such that
A16: k1 + 1 = k by NAT_1:6;
reconsider k1 = k1 as Element of NAT by ORDINAL1:def 12;
reconsider m = IC (Comput ((P +* (stop I)),(Initialize s),k1)) as Element of NAT ;
A17: card (stop (if=0 (a,k1,I))) = (card (stop I)) + 1 by A15, COMPOS_1:55;
m in dom (stop I) by A10, Def2;
then m < card (stop I) by AFINSQ_1:66;
then A18: m + 1 < card (stop (if=0 (a,k1,I))) by A17, XREAL_1:6;
IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),k)) = IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),1)),k1)) by A16, EXTPRO_1:4
.= m + 1 by A11, A14, A9, A8, Th45, UU ;
hence IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if=0 (a,k1,I))) by A18, AFINSQ_1:66; :: thesis:

end;

suppose k = 0 ; :: thesis:

hence IC (Comput ((P +* (stop (if=0 (a,k1,I)))),(Initialize s),k)) in dom (stop (if=0 (a,k1,I))) by A13, A4, EXTPRO_1:2; :: thesis:

end;