let s be State of SCMPDS; :: thesis: for k1, k2 being Integer
for a being Int_position holds
( (Exec (((a,k1) := k2),s)) . (IC ) = (IC s) + 1 & (Exec (((a,k1) := k2),s)) . (DataLoc ((s . a),k1)) = k2 & ( for b being Int_position st b <> DataLoc ((s . a),k1) holds
(Exec (((a,k1) := k2),s)) . b = s . b ) )
let k1, k2 be Integer; :: thesis: for a being Int_position holds
( (Exec (((a,k1) := k2),s)) . (IC ) = (IC s) + 1 & (Exec (((a,k1) := k2),s)) . (DataLoc ((s . a),k1)) = k2 & ( for b being Int_position st b <> DataLoc ((s . a),k1) holds
(Exec (((a,k1) := k2),s)) . b = s . b ) )
let a be Int_position; :: thesis: ( (Exec (((a,k1) := k2),s)) . (IC ) = (IC s) + 1 & (Exec (((a,k1) := k2),s)) . (DataLoc ((s . a),k1)) = k2 & ( for b being Int_position st b <> DataLoc ((s . a),k1) holds
(Exec (((a,k1) := k2),s)) . b = s . b ) )
reconsider S = s as SCM-State by CARD_3:107;
reconsider mk = a as Element of SCM-Data-Loc by AMI_2:def 16;
reconsider I = (a,k1) := k2 as Element of SCMPDS-Instr ;
set A2 = Address_Add (S,(I P31address),(I P32const));
set S1 = SCM-Chg (S,(Address_Add (S,(I P31address),(I P32const))),(I P33const));
reconsider i = 7 as Element of Segm 15 by NAT_1:44;
A1: I = [i,{},<*mk,k1,k2*>] ;
then A2: I P33const = k2 by SCMPDS_I:6;
A3: InsCode I = 7 ;
A4: Exec (((a,k1) := k2),s) = SCM-Exec-Res (I,S) by SCMPDS_1:def 23
.= SCM-Chg ((SCM-Chg (S,(Address_Add (S,(I P31address),(I P32const))),(I P33const))),((IC S) + 1)) by A3, SCMPDS_1:def 22 ;
hence (Exec (((a,k1) := k2),s)) . (IC ) = (IC s) + 1 by Th1, AMI_2:11; :: thesis: ( (Exec (((a,k1) := k2),s)) . (DataLoc ((s . a),k1)) = k2 & ( for b being Int_position st b <> DataLoc ((s . a),k1) holds
(Exec (((a,k1) := k2),s)) . b = s . b ) )
A5: ( I P31address = mk & I P32const = k1 ) by A1, SCMPDS_I:6;
hence (Exec (((a,k1) := k2),s)) . (DataLoc ((s . a),k1)) = (SCM-Chg (S,(Address_Add (S,(I P31address),(I P32const))),(I P33const))) . (Address_Add (S,(I P31address),(I P32const))) by A4, AMI_2:12
.= k2 by A2, AMI_2:15 ;
:: thesis: for b being Int_position st b <> DataLoc ((s . a),k1) holds
(Exec (((a,k1) := k2),s)) . b = s . b
let b be Int_position; :: thesis: ( b <> DataLoc ((s . a),k1) implies (Exec (((a,k1) := k2),s)) . b = s . b )
reconsider mn = b as Element of SCM-Data-Loc by AMI_2:def 16;
assume A6: b <> DataLoc ((s . a),k1) ; :: thesis: (Exec (((a,k1) := k2),s)) . b = s . b
thus (Exec (((a,k1) := k2),s)) . b = (SCM-Chg (S,(Address_Add (S,(I P31address),(I P32const))),(I P33const))) . mn by A4, AMI_2:12
.= s . b by A5, A6, AMI_2:16 ; :: thesis: verum