let P be Instruction-Sequence of SCMPDS; for s being State of SCMPDS
for I being halt-free parahalting shiftable Program of
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
(IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let s be State of SCMPDS; for I being halt-free parahalting shiftable Program of
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
(IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let I be halt-free parahalting shiftable Program of ; for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
(IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let a, b be Int_position; for k1 being Integer st s . (DataLoc ((s . a),k1)) < 0 holds
(IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
let k1 be Integer; ( s . (DataLoc ((s . a),k1)) < 0 implies (IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b )
assume A1:
s . (DataLoc ((s . a),k1)) < 0
; (IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
A2:
not b in dom (Start-At (((card I) + 1),SCMPDS))
by SCMPDS_4:18;
( I is_closed_on s,P & I is_halting_on s,P )
by Th11, Th12;
then
IExec ((if<0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))
by A1, Th101;
hence
(IExec ((if<0 (a,k1,I)),P,(Initialize s))) . b = (IExec (I,P,(Initialize s))) . b
by A2, FUNCT_4:11; verum