let i be Instruction of SCMPDS; :: thesis: for l being Element of NAT st ( for s being State of SCMPDS st IC s = l holds
(Exec (i,s)) . (IC SCMPDS) = succ (IC s) ) holds
NIC (i,l) = {(succ l)}
let l be Element of NAT ; :: thesis: ( ( for s being State of SCMPDS st IC s = l holds
(Exec (i,s)) . (IC SCMPDS) = succ (IC s) ) implies NIC (i,l) = {(succ l)} )
consider t being State of SCMPDS;
reconsider I = i as Instruction of SCMPDS ;
reconsider n = l as Element of NAT ;
assume A1: for s being State of SCMPDS st IC s = l holds
(Exec (i,s)) . (IC SCMPDS) = succ (IC s) ; :: thesis: NIC (i,l) = {(succ l)}

hereby :: according to TARSKI:def 3,XBOOLE_0:def 10 :: thesis: {(succ l)} c= NIC (i,l)

let x be set ; :: thesis: ( x in NIC (i,l) implies x in {(succ l)} )
assume x in NIC (i,l) ; :: thesis: x in {(succ l)}
then consider s being Element of product the Object-Kind of SCMPDS such that
A2: x = IC (Exec (i,s)) and
A3: IC s = l ;
x = succ l by A1, A2, A3;
hence x in {(succ l)} by TARSKI:def 1; :: thesis: verum

end;

reconsider il1 = l as Element of ObjectKind (IC SCMPDS) by COMPOS_1:def 6;
((IC SCMPDS),l) --> (il1,I) = ((IC SCMPDS) .--> il1) +* (l .--> I) by FUNCT_4:def 4;
then reconsider u = t +* (((IC SCMPDS),l) --> (il1,i)) as Element of product the Object-Kind of SCMPDS by PBOOLE:155;
A4: ( IC u = n & u . n = I ) by EXTPRO_1:26;
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {(succ l)} or x in NIC (i,l) )
assume x in {(succ l)} ; :: thesis: x in NIC (i,l)
then A5: x = succ l by TARSKI:def 1;
IC (Exec (I,u)) = succ l by A1, A4;
hence x in NIC (i,l) by A5, A4; :: thesis: verum