let il, i1 be Nat; :: thesis: NIC ((goto i1),il) = {i1}

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

let x be object ; :: thesis: ( x in {i1} iff x in { (IC (Exec ((goto i1),s))) where s is Element of product (the_Values_of SCM+FSA) : IC s = il } )

A1: now :: thesis: ( x = i1 implies x in { (IC (Exec ((goto i1),s))) where s is Element of product (the_Values_of SCM+FSA) : IC s = il } )

il in NAT by ORDINAL1:def 12;
then reconsider il1 = il as Element of Values (IC ) by MEMSTR_0:def 6;
reconsider n = il1 as Nat ;
set I = goto i1;
set t = the State of SCM+FSA;
set Q = the Instruction-Sequence of SCM+FSA;
assume A2: x = i1 ; :: thesis: x in { (IC (Exec ((goto i1),s))) where s is Element of product (the_Values_of SCM+FSA) : IC s = il }
reconsider u = the State of SCM+FSA +* ((IC ),il1) as Element of product (the_Values_of SCM+FSA) by CARD_3:107;
reconsider P = the Instruction-Sequence of SCM+FSA +* (il,(goto i1)) as Instruction-Sequence of SCM+FSA ;
IC in dom the State of SCM+FSA by MEMSTR_0:2;
then A3: IC u = n by FUNCT_7:31;
il in NAT by ORDINAL1:def 12;
then A4: P /. il = P . il by PBOOLE:143;
il in NAT by ORDINAL1:def 12;
then il in dom the Instruction-Sequence of SCM+FSA by PARTFUN1:def 2;
then A5: P . n = goto i1 by FUNCT_7:31;
then IC (Following (P,u)) = i1 by A3, A4, SCMFSA_2:69;
hence x in { (IC (Exec ((goto i1),s))) where s is Element of product (the_Values_of SCM+FSA) : IC s = il } by A2, A3, A5, A4; :: thesis: verum

end;

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

assume x in { (IC (Exec ((goto i1),s))) where s is Element of product (the_Values_of SCM+FSA) : IC s = il } ; :: thesis: x = i1
then ex s being Element of product (the_Values_of SCM+FSA) st
( x = IC (Exec ((goto i1),s)) & IC s = il ) ;
hence x = i1 by SCMFSA_2:69; :: thesis: verum

end;

hence ( x in {i1} iff x in { (IC (Exec ((goto i1),s))) where s is Element of product (the_Values_of SCM+FSA) : IC s = il } ) by A1, TARSKI:def 1; :: thesis: verum

end;

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