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

A1: now :: thesis: for k being Nat st S₁[k] holds
S₁[k + 1]

let k be 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 Th3
.= Comput (p,(Comput (p,s,i)),(k + 1)) by A2, Th3 ;
hence S₁[k + 1] ; :: thesis: verum

end;

A3: S₁[ 0 ] ;
thus for k being Nat holds S₁[k] from NAT_1:sch 2(A3, A1); :: thesis: verum