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 ) = (IC s) + 1 ) holds
NIC (i,l) = {(l + 1)}
let l be Element of NAT ; :: thesis: ( ( for s being State of SCMPDS st IC s = l holds
(Exec (i,s)) . (IC ) = (IC s) + 1 ) implies NIC (i,l) = {(l + 1)} )
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 ) = (IC s) + 1 ; :: thesis: NIC (i,l) = {(l + 1)}

hereby :: according to TARSKI:def 3,XBOOLE_0:def 10 :: thesis: {(l + 1)} c= NIC (i,l)

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

end;

set t = the l -started State of SCMPDS;
reconsider t = the l -started State of SCMPDS as Element of product (the_Values_of SCMPDS) by CARD_3:107;
A4: IC t = l by MEMSTR_0:def 11;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {(l + 1)} or x in NIC (i,l) )
assume x in {(l + 1)} ; :: thesis: x in NIC (i,l)
then A5: x = l + 1 by TARSKI:def 1;
IC (Exec (I,t)) = l + 1 by A1, A4;
hence x in NIC (i,l) by A5, A4; :: thesis: verum