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 s1, s2 being State of S st NPP s1 = NPP s2 holds
for P being the Instructions of b₁ -valued ManySortedSet of NAT
for k being Nat holds NPP (Comput (P,s1,k)) = NPP (Comput (P,s2,k))
let S be non empty stored-program IC-Ins-separated definite realistic halting Exec-preserving AMI-Struct of N; :: thesis: for s1, s2 being State of S st NPP s1 = NPP s2 holds
for P being the Instructions of S -valued ManySortedSet of NAT
for k being Nat holds NPP (Comput (P,s1,k)) = NPP (Comput (P,s2,k))
let s1, s2 be State of S; :: thesis: ( NPP s1 = NPP s2 implies for P being the Instructions of S -valued ManySortedSet of NAT
for k being Nat holds NPP (Comput (P,s1,k)) = NPP (Comput (P,s2,k)) )
assume Z: NPP s1 = NPP s2 ; :: thesis: for P being the Instructions of S -valued ManySortedSet of NAT
for k being Nat holds NPP (Comput (P,s1,k)) = NPP (Comput (P,s2,k))
let P be the Instructions of S -valued ManySortedSet of NAT ; :: thesis: for k being Nat holds NPP (Comput (P,s1,k)) = NPP (Comput (P,s2,k))
defpred S₁[ Nat] means NPP (Comput (P,s1,$1)) = NPP (Comput (P,s2,$1));
( Comput (P,s1,0) = s1 & Comput (P,s2,0) = s2 ) by EXTPRO_1:3;
then A: S₁[ 0 ] by Z;

B: now

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume ZZ: S₁[k] ; :: thesis: S₁[k + 1]
NPP (Comput (P,s1,(k + 1))) = NPP (Following (P,(Comput (P,s1,k)))) by EXTPRO_1:4
.= NPP (Following (P,(Comput (P,s2,k)))) by ZZ, ThX
.= NPP (Comput (P,s2,(k + 1))) by EXTPRO_1:4 ;
hence S₁[k + 1] ; :: thesis: verum

end;

thus for k being Nat holds S₁[k] from NAT_1:sch 2(A, B); :: thesis: verum