let R be Ring; :: thesis: for a, b being Data-Location of R
for s being State of (SCM R) holds
( (Exec ((AddTo (a,b)),s)) . () = (IC s) + 1 & (Exec ((AddTo (a,b)),s)) . a = (s . a) + (s . b) & ( for c being Data-Location of R st c <> a holds
(Exec ((AddTo (a,b)),s)) . c = s . c ) )

let a, b be Data-Location of R; :: thesis: for s being State of (SCM R) holds
( (Exec ((AddTo (a,b)),s)) . () = (IC s) + 1 & (Exec ((AddTo (a,b)),s)) . a = (s . a) + (s . b) & ( for c being Data-Location of R st c <> a holds
(Exec ((AddTo (a,b)),s)) . c = s . c ) )

let s be State of (SCM R); :: thesis: ( (Exec ((AddTo (a,b)),s)) . () = (IC s) + 1 & (Exec ((AddTo (a,b)),s)) . a = (s . a) + (s . b) & ( for c being Data-Location of R st c <> a holds
(Exec ((AddTo (a,b)),s)) . c = s . c ) )

A1: a is Element of Data-Locations by Th1;
A2: the_Values_of (SCM R) = () * SCM-OK by Lm1;
reconsider S = s as SCM-State of R by ;
reconsider I = AddTo (a,b) as Element of SCM-Instr R by Def1;
set S1 = SCM-Chg (S,(),((S . ()) + (S . ())));
reconsider i = 2 as Element of Segm 8 by NAT_1:44;
A3: IC s = IC S by Def1;
A4: b is Element of Data-Locations by Th1;
A5: Exec ((AddTo (a,b)),s) = SCM-Exec-Res (I,S) by Th10
.= SCM-Chg ((SCM-Chg (S,(),((S . ()) + (S . ())))),((IC S) + 1)) by ;
A6: I = [i,{},<*a,b*>] ;
then A7: I address_1 = a by ;
A8: I address_2 = b by ;
thus (Exec ((AddTo (a,b)),s)) . () = (Exec ((AddTo (a,b)),s)) . NAT by Def1
.= (IC s) + 1 by ; :: thesis: ( (Exec ((AddTo (a,b)),s)) . a = (s . a) + (s . b) & ( for c being Data-Location of R st c <> a holds
(Exec ((AddTo (a,b)),s)) . c = s . c ) )

thus (Exec ((AddTo (a,b)),s)) . a = (SCM-Chg (S,(),((S . ()) + (S . ())))) . a by
.= (s . a) + (s . b) by ; :: thesis: for c being Data-Location of R st c <> a holds
(Exec ((AddTo (a,b)),s)) . c = s . c

let c be Data-Location of R; :: thesis: ( c <> a implies (Exec ((AddTo (a,b)),s)) . c = s . c )
assume A9: c <> a ; :: thesis: (Exec ((AddTo (a,b)),s)) . c = s . c
A10: c is Element of Data-Locations by Th1;
hence (Exec ((AddTo (a,b)),s)) . c = (SCM-Chg (S,(),((S . ()) + (S . ())))) . c by
.= s . c by ;
:: thesis: verum