let s be State of SCMPDS; :: thesis: for k1, k2 being Integer
for a, b being Int_position holds
( (Exec ((MultBy (a,k1,b,k2)),s)) . (IC ) = succ (IC s) & (Exec ((MultBy (a,k1,b,k2)),s)) . (DataLoc ((s . a),k1)) = (s . (DataLoc ((s . a),k1))) * (s . (DataLoc ((s . b),k2))) & ( for c being Int_position st c <> DataLoc ((s . a),k1) holds
(Exec ((MultBy (a,k1,b,k2)),s)) . c = s . c ) )
let k1, k2 be Integer; :: thesis: for a, b being Int_position holds
( (Exec ((MultBy (a,k1,b,k2)),s)) . (IC ) = succ (IC s) & (Exec ((MultBy (a,k1,b,k2)),s)) . (DataLoc ((s . a),k1)) = (s . (DataLoc ((s . a),k1))) * (s . (DataLoc ((s . b),k2))) & ( for c being Int_position st c <> DataLoc ((s . a),k1) holds
(Exec ((MultBy (a,k1,b,k2)),s)) . c = s . c ) )
let a, b be Int_position; :: thesis: ( (Exec ((MultBy (a,k1,b,k2)),s)) . (IC ) = succ (IC s) & (Exec ((MultBy (a,k1,b,k2)),s)) . (DataLoc ((s . a),k1)) = (s . (DataLoc ((s . a),k1))) * (s . (DataLoc ((s . b),k2))) & ( for c being Int_position st c <> DataLoc ((s . a),k1) holds
(Exec ((MultBy (a,k1,b,k2)),s)) . c = s . c ) )
reconsider S = s as SCM-State by CARD_3:107;
reconsider da = a, db = b as Element of SCM-Data-Loc by AMI_2:def 16;
reconsider I = MultBy (a,k1,b,k2) as Element of SCMPDS-Instr ;
set A2 = Address_Add (S,(I P41address),(I P43const));
set A4 = Address_Add (S,(I P42address),(I P44const));
set S1 = SCM-Chg (S,(Address_Add (S,(I P41address),(I P43const))),((S . (Address_Add (S,(I P41address),(I P43const)))) * (S . (Address_Add (S,(I P42address),(I P44const))))));
reconsider i = 11 as Element of Segm 15 by NAT_1:44;
A1: I = [i,{},<*da,db,k1,k2*>] ;
then A2: ( I P42address = db & I P44const = k2 ) by SCMPDS_I:7;
A3: InsCode I = 11 by RECDEF_2:def 1;
A4: Exec ((MultBy (a,k1,b,k2)),s) = SCM-Exec-Res (I,S) by SCMPDS_1:def 23
.= SCM-Chg ((SCM-Chg (S,(Address_Add (S,(I P41address),(I P43const))),((S . (Address_Add (S,(I P41address),(I P43const)))) * (S . (Address_Add (S,(I P42address),(I P44const))))))),(succ (IC S))) by A3, SCMPDS_1:def 22 ;
hence (Exec ((MultBy (a,k1,b,k2)),s)) . (IC ) = succ (IC s) by Th2, AMI_2:11; :: thesis: ( (Exec ((MultBy (a,k1,b,k2)),s)) . (DataLoc ((s . a),k1)) = (s . (DataLoc ((s . a),k1))) * (s . (DataLoc ((s . b),k2))) & ( for c being Int_position st c <> DataLoc ((s . a),k1) holds
(Exec ((MultBy (a,k1,b,k2)),s)) . c = s . c ) )
A5: ( I P41address = da & I P43const = k1 ) by A1, SCMPDS_I:7;
hence (Exec ((MultBy (a,k1,b,k2)),s)) . (DataLoc ((s . a),k1)) = (SCM-Chg (S,(Address_Add (S,(I P41address),(I P43const))),((S . (Address_Add (S,(I P41address),(I P43const)))) * (S . (Address_Add (S,(I P42address),(I P44const))))))) . (Address_Add (S,(I P41address),(I P43const))) by A4, AMI_2:12
.= (s . (DataLoc ((s . a),k1))) * (s . (DataLoc ((s . b),k2))) by A5, A2, AMI_2:15 ;
:: thesis: for c being Int_position st c <> DataLoc ((s . a),k1) holds
(Exec ((MultBy (a,k1,b,k2)),s)) . c = s . c
let c be Int_position; :: thesis: ( c <> DataLoc ((s . a),k1) implies (Exec ((MultBy (a,k1,b,k2)),s)) . c = s . c )
reconsider mn = c as Element of SCM-Data-Loc by AMI_2:def 16;
assume A6: c <> DataLoc ((s . a),k1) ; :: thesis: (Exec ((MultBy (a,k1,b,k2)),s)) . c = s . c
thus (Exec ((MultBy (a,k1,b,k2)),s)) . c = (SCM-Chg (S,(Address_Add (S,(I P41address),(I P43const))),((S . (Address_Add (S,(I P41address),(I P43const)))) * (S . (Address_Add (S,(I P42address),(I P44const))))))) . mn by A4, AMI_2:12
.= s . c by A5, A6, AMI_2:16 ; :: thesis: verum