let R be Ring; :: thesis: for il, i1 being Nat holds NIC ((goto (i1,R)),il) = {i1}
let il, i1 be Nat; :: thesis: NIC ((goto (i1,R)),il) = {i1}

now :: thesis: for x being object holds
( x in {i1} iff x in { (IC (Exec ((goto (i1,R)),s))) where s is Element of product (the_Values_of (SCM R)) : IC s = il } )

let x be object ; :: thesis: ( x in {i1} iff x in { (IC (Exec ((goto (i1,R)),s))) where s is Element of product (the_Values_of (SCM R)) : IC s = il } )
A1: il in NAT by ORDINAL1:def 12;

A2: now :: thesis: ( x = i1 implies x in { (IC (Exec ((goto (i1,R)),s))) where s is Element of product (the_Values_of (SCM R)) : IC s = il } )

reconsider il1 = il as Element of Values (IC ) by MEMSTR_0:def 6, A1;
set I = goto (i1,R);
set t = the State of (SCM R);
set Q = the Instruction-Sequence of (SCM R);
assume A3: x = i1 ; :: thesis: x in { (IC (Exec ((goto (i1,R)),s))) where s is Element of product (the_Values_of (SCM R)) : IC s = il }
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,(goto (i1,R))) as Instruction-Sequence of (SCM R) ;
A4: P /. il = P . il by PBOOLE:143, A1;
IC in dom the State of (SCM R) by MEMSTR_0:2;
then A5: IC u = il by FUNCT_7:31;
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 = goto (i1,R) by FUNCT_7:31;
then IC (Following (P,u)) = i1 by A5, A4, SCMRING2:15;
hence x in { (IC (Exec ((goto (i1,R)),s))) where s is Element of product (the_Values_of (SCM R)) : IC s = il } by A3, A4, A5, A6; :: thesis: verum

end;

now :: thesis: ( x in { (IC (Exec ((goto (i1,R)),s))) where s is Element of product (the_Values_of (SCM R)) : IC s = il } implies x = i1 )

assume x in { (IC (Exec ((goto (i1,R)),s))) where s is Element of product (the_Values_of (SCM R)) : IC s = il } ; :: thesis: x = i1
then ex s being Element of product (the_Values_of (SCM R)) st
( x = IC (Exec ((goto (i1,R)),s)) & IC s = il ) ;
hence x = i1 by SCMRING2:15; :: thesis: verum

end;

hence ( x in {i1} iff x in { (IC (Exec ((goto (i1,R)),s))) where s is Element of product (the_Values_of (SCM R)) : IC s = il } ) by A2, TARSKI:def 1; :: thesis: verum

end;

hence NIC ((goto (i1,R)),il) = {i1} by TARSKI:2; :: thesis: verum