let il be Element of NAT ; :: thesis: NIC ((halt SCM+FSA),il) = {il}

now

let x be set ; :: thesis: ( x in {il} iff x in { (IC (Exec ((halt SCM+FSA),s))) where s is Element of product the Object-Kind of SCM+FSA : IC s = il } )

A1: now

reconsider il1 = il as Element of ObjectKind (IC SCM+FSA) by COMPOS_1:def 6;
reconsider I = halt SCM+FSA as Element of the Object-Kind of SCM+FSA . il by COMPOS_1:def 8;
reconsider n = il1 as Element of NAT ;
consider t being State of SCM+FSA;
assume A2: x = il ; :: thesis: x in { (IC (Exec ((halt SCM+FSA),s))) where s is Element of product the Object-Kind of SCM+FSA : IC s = il }
reconsider p = ((IC SCM+FSA),il) --> (il1,I) as PartState of SCM+FSA by COMPOS_1:37;
reconsider u = t +* p as Element of product the Object-Kind of SCM+FSA by PBOOLE:155;
A3: IC u = n by EXTPRO_1:26;
dom (((IC SCM+FSA),il) --> (il1,I)) = {(IC SCM+FSA),il} by FUNCT_4:65;
then A4: IC SCM+FSA in dom (((IC SCM+FSA),il) --> (il1,I)) by TARSKI:def 2;
IC (Exec ((halt SCM+FSA),u)) = u . (IC SCM+FSA) by EXTPRO_1:def 3
.= (((IC SCM+FSA),il) --> (il1,I)) . (IC SCM+FSA) by A4, FUNCT_4:14
.= il by COMPOS_1:3, FUNCT_4:66 ;
hence x in { (IC (Exec ((halt SCM+FSA),s))) where s is Element of product the Object-Kind of SCM+FSA : IC s = il } by A2, A3; :: thesis: verum

end;

now

assume x in { (IC (Exec ((halt SCM+FSA),s))) where s is Element of product the Object-Kind of SCM+FSA : IC s = il } ; :: thesis: x = il
then ex s being Element of product the Object-Kind of SCM+FSA st
( x = IC (Exec ((halt SCM+FSA),s)) & IC s = il ) ;
hence x = il by EXTPRO_1:def 3; :: thesis: verum

end;

hence ( x in {il} iff x in { (IC (Exec ((halt SCM+FSA),s))) where s is Element of product the Object-Kind of SCM+FSA : IC s = il } ) by A1, TARSKI:def 1; :: thesis: verum

end;

hence NIC ((halt SCM+FSA),il) = {il} by TARSKI:2; :: thesis: verum