let P be Instruction-Sequence of SCMPDS; for s being State of SCMPDS
for I being 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 = s . b
let s be State of SCMPDS; for I being 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 = s . b
let I be 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 = 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 = s . b
let k1 be Integer; ( s . (DataLoc ((s . a),k1)) <= 0 implies (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b = s . b )
assume
s . (DataLoc ((s . a),k1)) <= 0
; (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b = s . b
then A1:
IExec ((if>0 (a,k1,I)),P,(Initialize s)) = s +* (Start-At (((card I) + 1),SCMPDS))
by Th72;
not b in dom (Start-At (((card I) + 1),SCMPDS))
by SCMPDS_4:18;
hence
(IExec ((if>0 (a,k1,I)),P,(Initialize s))) . b = s . b
by A1, FUNCT_4:11; verum