let R be Ring; :: thesis: for a being Data-Location of R

for il, i1 being Nat holds

( i1 in NIC ((a =0_goto i1),il) & NIC ((a =0_goto i1),il) c= {i1,(il + 1)} )

let a be Data-Location of R; :: thesis: for il, i1 being Nat holds

( i1 in NIC ((a =0_goto i1),il) & NIC ((a =0_goto i1),il) c= {i1,(il + 1)} )

let il, i1 be Nat; :: thesis: ( i1 in NIC ((a =0_goto i1),il) & NIC ((a =0_goto i1),il) c= {i1,(il + 1)} )

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;

A1: il in NAT by ORDINAL1:def 12;

reconsider il1 = il as Element of Values (IC ) by MEMSTR_0:def 6, A1;

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;

A2: IC in dom the State of (SCM R) by MEMSTR_0:2;

IC <> a by Th2;

then not IC in dom (a .--> 0R) by TARSKI:def 1;

then A4: IC v = IC u by FUNCT_4:11

.= il by A2, FUNCT_7:31 ;

A5: P /. il = P . il by PBOOLE:143, A1;

il in NAT by ORDINAL1:def 12;

then il in dom the Instruction-Sequence of (SCM R) by PARTFUN1:def 2;

then A6: P . il = a =0_goto i1 by FUNCT_7:31;

a in dom (a .--> 0R) by 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 A4, A6, A5, SCMRING2:16;

hence i1 in NIC ((a =0_goto i1),il) by A4, A6, A5; :: thesis: NIC ((a =0_goto i1),il) c= {i1,(il + 1)}

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in NIC ((a =0_goto i1),il) or x in {i1,(il + 1)} )

assume x in NIC ((a =0_goto i1),il) ; :: thesis: x in {i1,(il + 1)}

then consider s being Element of product (the_Values_of (SCM R)) such that

A7: ( x = IC (Exec ((a =0_goto i1),s)) & IC s = il ) ;

for il, i1 being Nat holds

( i1 in NIC ((a =0_goto i1),il) & NIC ((a =0_goto i1),il) c= {i1,(il + 1)} )

let a be Data-Location of R; :: thesis: for il, i1 being Nat holds

( i1 in NIC ((a =0_goto i1),il) & NIC ((a =0_goto i1),il) c= {i1,(il + 1)} )

let il, i1 be Nat; :: thesis: ( i1 in NIC ((a =0_goto i1),il) & NIC ((a =0_goto i1),il) c= {i1,(il + 1)} )

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;

A1: il in NAT by ORDINAL1:def 12;

reconsider il1 = il as Element of Values (IC ) by MEMSTR_0:def 6, A1;

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;

A2: IC in dom the State of (SCM R) by MEMSTR_0:2;

IC <> a by Th2;

then not IC in dom (a .--> 0R) by TARSKI:def 1;

then A4: IC v = IC u by FUNCT_4:11

.= il by A2, FUNCT_7:31 ;

A5: P /. il = P . il by PBOOLE:143, A1;

il in NAT by ORDINAL1:def 12;

then il in dom the Instruction-Sequence of (SCM R) by PARTFUN1:def 2;

then A6: P . il = a =0_goto i1 by FUNCT_7:31;

a in dom (a .--> 0R) by 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 A4, A6, A5, SCMRING2:16;

hence i1 in NIC ((a =0_goto i1),il) by A4, A6, A5; :: thesis: NIC ((a =0_goto i1),il) c= {i1,(il + 1)}

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in NIC ((a =0_goto i1),il) or x in {i1,(il + 1)} )

assume x in NIC ((a =0_goto i1),il) ; :: thesis: x in {i1,(il + 1)}

then consider s being Element of product (the_Values_of (SCM R)) such that

A7: ( x = IC (Exec ((a =0_goto i1),s)) & IC s = il ) ;