let a, b be Data-Location; :: thesis: for s being State of SCM holds
( (Exec ((MultBy (a,b)),s)) . (IC ) = (IC s) + 1 & (Exec ((MultBy (a,b)),s)) . a = (s . a) * (s . b) & ( for c being Data-Location st c <> a holds
(Exec ((MultBy (a,b)),s)) . c = s . c ) )
let s be State of SCM; :: thesis: ( (Exec ((MultBy (a,b)),s)) . (IC ) = (IC s) + 1 & (Exec ((MultBy (a,b)),s)) . a = (s . a) * (s . b) & ( for c being Data-Location st c <> a holds
(Exec ((MultBy (a,b)),s)) . c = s . c ) )
reconsider S = s as SCM-State by CARD_3:107;
reconsider mk = a, ml = b as Element of SCM-Data-Loc by AMI_2:def 16;
reconsider I = MultBy (a,b) as Element of SCM-Instr ;
set S1 = SCM-Chg (S,(I address_1),((S . (I address_1)) * (S . (I address_2))));
reconsider i = 4 as Element of Segm 9 by NAT_1:44;
A1: I = [i,{},<*mk,ml*>] ;
then A2: I address_1 = mk by SCM_INST:5;
A3: I address_2 = ml by A1, SCM_INST:5;
A4: Exec ((MultBy (a,b)),s) = SCM-Exec-Res (I,S) by AMI_2:def 15
.= SCM-Chg ((SCM-Chg (S,(I address_1),((S . (I address_1)) * (S . (I address_2))))),((IC S) + 1)) by A1, AMI_2:def 14 ;
hence (Exec ((MultBy (a,b)),s)) . (IC ) = (IC s) + 1 by Th1, AMI_2:11; :: thesis: ( (Exec ((MultBy (a,b)),s)) . a = (s . a) * (s . b) & ( for c being Data-Location st c <> a holds
(Exec ((MultBy (a,b)),s)) . c = s . c ) )
thus (Exec ((MultBy (a,b)),s)) . a = (SCM-Chg (S,(I address_1),((S . (I address_1)) * (S . (I address_2))))) . mk by A4, AMI_2:12
.= (s . a) * (s . b) by A2, A3, AMI_2:15 ; :: thesis: for c being Data-Location st c <> a holds
(Exec ((MultBy (a,b)),s)) . c = s . c
let c be Data-Location; :: thesis: ( c <> a implies (Exec ((MultBy (a,b)),s)) . c = s . c )
reconsider mn = c as Element of SCM-Data-Loc by AMI_2:def 16;
assume A5: c <> a ; :: thesis: (Exec ((MultBy (a,b)),s)) . c = s . c
thus (Exec ((MultBy (a,b)),s)) . c = (SCM-Chg (S,(I address_1),((S . (I address_1)) * (S . (I address_2))))) . mn by A4, AMI_2:12
.= s . c by A2, A5, AMI_2:16 ; :: thesis: verum