let P be the Instructions of SCMPDS -valued ManySortedSet of NAT ; :: thesis: for s being State of SCMPDS
for I being Program of SCMPDS
for J being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if=0 (a,k1,I,J)),P,s)) . b = (IExec (J,P,s)) . b
let s be State of SCMPDS; :: thesis: for I being Program of SCMPDS
for J being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if=0 (a,k1,I,J)),P,s)) . b = (IExec (J,P,s)) . b
let I be Program of SCMPDS; :: thesis: for J being halt-free parahalting shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if=0 (a,k1,I,J)),P,s)) . b = (IExec (J,P,s)) . b
let J be halt-free parahalting shiftable Program of SCMPDS; :: thesis: for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if=0 (a,k1,I,J)),P,s)) . b = (IExec (J,P,s)) . b
let a, b be Int_position ; :: thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if=0 (a,k1,I,J)),P,s)) . b = (IExec (J,P,s)) . b
let k1 be Integer; :: thesis: ( s . (DataLoc ((s . a),k1)) <> 0 implies (IExec ((if=0 (a,k1,I,J)),P,s)) . b = (IExec (J,P,s)) . b )
assume A1: s . (DataLoc ((s . a),k1)) <> 0 ; :: thesis: (IExec ((if=0 (a,k1,I,J)),P,s)) . b = (IExec (J,P,s)) . b
A2: not b in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by SCMPDS_4:59;
( J is_closed_on s,P & J is_halting_on s,P ) by Th34, Th35;
then IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A1, Th55;
hence (IExec ((if=0 (a,k1,I,J)),P,s)) . b = (IExec (J,P,s)) . b by A2, FUNCT_4:12; :: thesis: verum