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 b_{2} -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 b_{1} -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 S_{1}[ Nat] means Comput (p,s,(i + $1)) = Comput (p,(Comput (p,s,i)),$1);

_{1}[ 0 ]
;

thus for k being Nat holds S_{1}[k]
from NAT_1:sch 2(A3, A1); :: thesis: verum

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 b

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 b

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 S

A1: now :: thesis: for k being Nat st S_{1}[k] holds

S_{1}[k + 1]

A3:
SS

let k be Nat; :: thesis: ( S_{1}[k] implies S_{1}[k + 1] )

assume A2: S_{1}[k]
; :: thesis: S_{1}[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 S_{1}[k + 1]
; :: thesis: verum

end;assume A2: S

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 S

thus for k being Nat holds S