let R be non trivial Ring; :: thesis: for a being Data-Location of R
for il, i1 being Element of NAT holds NIC ((a =0_goto i1),il) = {i1,(il + 1)}

let a be Data-Location of R; :: thesis: for il, i1 being Element of NAT holds NIC ((a =0_goto i1),il) = {i1,(il + 1)}
let il, i1 be Element of NAT ; :: thesis: NIC ((a =0_goto i1),il) = {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: Values a = (() * SCM-OK) . a9 by SCMRING2:24
.= the carrier of R by ;
reconsider il1 = il as Element of Values () by MEMSTR_0:def 6;
thus NIC ((a =0_goto i1),il) c= {i1,(il + 1)} by Th31; :: according to XBOOLE_0:def 10 :: thesis: {i1,(il + 1)} c= NIC ((a =0_goto i1),il)
reconsider u = the State of (SCM R) +* ((),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) ;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {i1,(il + 1)} or x in NIC ((a =0_goto i1),il) )
A2: IC <> a by Th2;
A3: IC in dom the State of (SCM R) by MEMSTR_0:2;
assume A4: x in {i1,(il + 1)} ; :: thesis: x in NIC ((a =0_goto i1),il)
per cases ( x = i1 or x = il + 1 ) by ;
suppose A5: x = i1 ; :: thesis: x in NIC ((a =0_goto i1),il)
reconsider 0R = 0. R as Element of Values a by A1;
reconsider v = u +* (a .--> 0R) as Element of product (the_Values_of (SCM R)) by CARD_3:107;
not IC in dom (a .--> 0R) by ;
then A7: IC v = IC u by FUNCT_4:11
.= il by ;
A8: 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 A9: 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 ;
hence x in NIC ((a =0_goto i1),il) by A5, A7, A8, A9; :: thesis: verum
end;
suppose A10: x = il + 1 ; :: thesis: x in NIC ((a =0_goto i1),il)
consider e being Element of R such that
A11: e <> 0. R by STRUCT_0:def 18;
reconsider E = e as Element of Values a by A1;
reconsider v = u +* (a .--> E) as Element of product (the_Values_of (SCM R)) by CARD_3:107;
not IC in dom (a .--> E) by ;
then A13: IC v = IC u by FUNCT_4:11
.= il by ;
A14: 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 A15: P . il = a =0_goto i1 by FUNCT_7:31;
a in dom (a .--> E) by TARSKI:def 1;
then v . a = (a .--> E) . a by FUNCT_4:13
.= E by FUNCOP_1:72 ;
then IC (Following (P,v)) = il + 1 by ;
hence x in NIC ((a =0_goto i1),il) by A10, A13, A14, A15; :: thesis: verum
end;
end;