let i be Element of NAT ; :: thesis: for N being non empty with_non-empty_elements set
for S being non empty stored-program IC-Ins-separated definite AMI-Struct of N
for s being State of S
for p being NAT -defined the Instructions of b₂ -valued Function
for k being Element of NAT holds Comput p,s,(i + k) = Comput p,(Comput p,s,i),k
let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty stored-program IC-Ins-separated definite AMI-Struct of N
for s being State of S
for p being NAT -defined the Instructions of b₁ -valued Function
for k being Element of NAT holds Comput p,s,(i + k) = Comput p,(Comput p,s,i),k
let S be non empty stored-program IC-Ins-separated definite AMI-Struct of N; :: thesis: for s being State of S
for p being NAT -defined the Instructions of S -valued Function
for k being Element of NAT holds Comput p,s,(i + k) = Comput p,(Comput p,s,i),k
let s be State of S; :: thesis: for p being NAT -defined the Instructions of S -valued Function
for k being Element of NAT holds Comput p,s,(i + k) = Comput p,(Comput p,s,i),k
let p be NAT -defined the Instructions of S -valued Function; :: thesis: for k being Element of NAT holds Comput p,s,(i + k) = Comput p,(Comput p,s,i),k
defpred S₁[ Element of NAT ] means Comput p,s,(i + $1) = Comput p,(Comput p,s,i),$1;

A1: now

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A2: S₁[k] ; :: thesis: S₁[k + 1]
Comput p,s,(i + (k + 1)) = Comput p,s,((i + k) + 1)
.= Following p,(Comput p,s,(i + k)) by Th14
.= Comput p,(Comput p,s,i),(k + 1) by A2, Th14 ;
hence S₁[k + 1] ; :: thesis: verum

end;

A3: S₁[ 0 ] by Th13;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A3, A1); :: thesis: verum