let N be with_non-empty_elements set ; :: thesis: for i being Element of NAT
for S being non empty stored-program IC-Ins-separated definite AMI-Struct of N
for s being State of S
for k being Element of NAT holds Computation s,(i + k) = Computation (Computation s,i),k
let i be Element of NAT ; :: thesis: for S being non empty stored-program IC-Ins-separated definite AMI-Struct of N
for s being State of S
for k being Element of NAT holds Computation s,(i + k) = Computation (Computation 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 k being Element of NAT holds Computation s,(i + k) = Computation (Computation s,i),k
let s be State of S; :: thesis: for k being Element of NAT holds Computation s,(i + k) = Computation (Computation s,i),k
defpred S₁[ Element of NAT ] means Computation s,(i + $1) = Computation (Computation 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]
Computation s,(i + (k + 1)) = Computation s,((i + k) + 1)
.= Following (Computation s,(i + k)) by Th14
.= Computation (Computation 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