let R be Ring; :: thesis: for a being Data-Location of R
for i1, il being Element of NAT holds
( i1 in NIC ((a =0_goto i1),il) & NIC ((a =0_goto i1),il) c= {i1,(succ il)} )
let a be Data-Location of R; :: thesis: for i1, il being Element of NAT holds
( i1 in NIC ((a =0_goto i1),il) & NIC ((a =0_goto i1),il) c= {i1,(succ il)} )
let i1, il be Element of NAT ; :: thesis: ( i1 in NIC ((a =0_goto i1),il) & NIC ((a =0_goto i1),il) c= {i1,(succ il)} )
set t = the State of (SCM R);
set Q = the Instruction-Sequence of (SCM R);
set I = a =0_goto i1;
reconsider a9 = a as Element of Data-Locations by SCMRING2:1;
reconsider il1 = il as Element of Values (IC ) by MEMSTR_0:def 6;
Values a = ((SCM-VAL R) * SCM-OK) . a9 by SCMRING2:24
.= the carrier of R by AMI_3:27, SCMRING1:4 ;
then reconsider 0R = 0. R as Element of Values a ;
reconsider u = the State of (SCM R) +* ((IC ),il1) as Element of product (the_Values_of (SCM R)) by CARD_3:107;
reconsider P = the Instruction-Sequence of (SCM R) +* (il,(a =0_goto i1)) as Instruction-Sequence of (SCM R) ;
reconsider v = u +* (a .--> 0R) as Element of product (the_Values_of (SCM R)) by CARD_3:107;
A1: IC in dom the State of (SCM R) by MEMSTR_0:2;
A2: dom (a .--> 0R) = {a} by FUNCOP_1:13;
IC <> a by Th2;
then not IC in dom (a .--> 0R) by A2, TARSKI:def 1;
then A3: IC v = IC u by FUNCT_4:11
.= il by A1, FUNCT_7:31 ;
A4: P /. il = P . il by PBOOLE:143;
il in NAT ;
then il in dom the Instruction-Sequence of (SCM R) by PARTFUN1:def 2;
then A5: P . il = a =0_goto i1 by FUNCT_7:31;
a in dom (a .--> 0R) by A2, TARSKI:def 1;
then v . a = (a .--> 0R) . a by FUNCT_4:13
.= 0. R by FUNCOP_1:72 ;
then IC (Following (P,v)) = i1 by A3, A5, A4, SCMRING2:16;
hence i1 in NIC ((a =0_goto i1),il) by A3, A5, A4; :: thesis: NIC ((a =0_goto i1),il) c= {i1,(succ il)}
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in NIC ((a =0_goto i1),il) or x in {i1,(succ il)} )
assume x in NIC ((a =0_goto i1),il) ; :: thesis: x in {i1,(succ il)}
then consider s being Element of product (the_Values_of (SCM R)) such that
A6: ( x = IC (Exec ((a =0_goto i1),s)) & IC s = il ) ;

per cases ( s . a = 0. R or s . a <> 0. R ) ;

suppose s . a = 0. R ; :: thesis: x in {i1,(succ il)}

then x = i1 by A6, SCMRING2:16;
hence x in {i1,(succ il)} by TARSKI:def 2; :: thesis: verum

end;

suppose s . a <> 0. R ; :: thesis: x in {i1,(succ il)}

then x = succ il by A6, SCMRING2:16;
hence x in {i1,(succ il)} by TARSKI:def 2; :: thesis: verum

end;

end;