let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty stored-program IC-Ins-separated definite realistic halting Exec-preserving AMI-Struct of N
for P being the Instructions of b₁ -valued ManySortedSet of NAT
for s being State of S st NPP s = NPP (Following (P,s)) holds
for n being Element of NAT holds NPP (Comput (P,s,n)) = NPP s
let S be non empty stored-program IC-Ins-separated definite realistic halting Exec-preserving AMI-Struct of N; :: thesis: for P being the Instructions of S -valued ManySortedSet of NAT
for s being State of S st NPP s = NPP (Following (P,s)) holds
for n being Element of NAT holds NPP (Comput (P,s,n)) = NPP s
let P be the Instructions of S -valued ManySortedSet of NAT ; :: thesis: for s being State of S st NPP s = NPP (Following (P,s)) holds
for n being Element of NAT holds NPP (Comput (P,s,n)) = NPP s
let s be State of S; :: thesis: ( NPP s = NPP (Following (P,s)) implies for n being Element of NAT holds NPP (Comput (P,s,n)) = NPP s )
defpred S₁[ Element of NAT ] means NPP (Comput (P,s,$1)) = NPP s;
assume Z: NPP s = NPP (Following (P,s)) ; :: thesis: for n being Element of NAT holds NPP (Comput (P,s,n)) = NPP s

A1: now

let n be Element of NAT ; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume ZZ: S₁[n] ; :: thesis: S₁[n + 1]
NPP (Comput (P,s,(n + 1))) = NPP (Following (P,(Comput (P,s,n)))) by EXTPRO_1:4
.= NPP s by Z, ThX, ZZ ;
hence S₁[n + 1] ; :: thesis: verum

end;

A2: S₁[ 0 ] by EXTPRO_1:3;
thus for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A2, A1); :: thesis: verum