let R be Ring; :: thesis: for a, c being Data-Location of R
for i1 being Nat
for s being State of (SCM R) holds
( ( s . a = 0. R implies (Exec ((a =0_goto i1),s)) . () = i1 ) & ( s . a <> 0. R implies (Exec ((a =0_goto i1),s)) . () = (IC s) + 1 ) & (Exec ((a =0_goto i1),s)) . c = s . c )

let a, c be Data-Location of R; :: thesis: for i1 being Nat
for s being State of (SCM R) holds
( ( s . a = 0. R implies (Exec ((a =0_goto i1),s)) . () = i1 ) & ( s . a <> 0. R implies (Exec ((a =0_goto i1),s)) . () = (IC s) + 1 ) & (Exec ((a =0_goto i1),s)) . c = s . c )

let i1 be Nat; :: thesis: for s being State of (SCM R) holds
( ( s . a = 0. R implies (Exec ((a =0_goto i1),s)) . () = i1 ) & ( s . a <> 0. R implies (Exec ((a =0_goto i1),s)) . () = (IC s) + 1 ) & (Exec ((a =0_goto i1),s)) . c = s . c )

let s be State of (SCM R); :: thesis: ( ( s . a = 0. R implies (Exec ((a =0_goto i1),s)) . () = i1 ) & ( s . a <> 0. R implies (Exec ((a =0_goto i1),s)) . () = (IC s) + 1 ) & (Exec ((a =0_goto i1),s)) . c = s . c )
A1: the_Values_of (SCM R) = () * SCM-OK by Lm1;
reconsider S = s as SCM-State of R by ;
reconsider I = a =0_goto i1 as Element of SCM-Instr R by Def1;
reconsider i = 7 as Element of Segm 8 by NAT_1:44;
A2: ( a is Element of Data-Locations & i1 is Element of NAT ) by ;
A3: Exec ((a =0_goto i1),s) = SCM-Exec-Res (I,S) by Th10
.= SCM-Chg (S,(IFEQ ((S . ()),(0. R),(),((IC S) + 1)))) by ;
A4: I = [i,<*i1*>,<*a*>] ;
thus ( s . a = 0. R implies (Exec ((a =0_goto i1),s)) . () = i1 ) :: thesis: ( ( s . a <> 0. R implies (Exec ((a =0_goto i1),s)) . () = (IC s) + 1 ) & (Exec ((a =0_goto i1),s)) . c = s . c )
proof
assume s . a = 0. R ; :: thesis: (Exec ((a =0_goto i1),s)) . () = i1
then A5: S . () = 0. R by ;
thus (Exec ((a =0_goto i1),s)) . () = (Exec ((a =0_goto i1),s)) . NAT by Def1
.= IFEQ ((S . ()),(0. R),(),((IC S) + 1)) by