let k 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 steady-programmed AMI-Struct of N
for P being the Instructions of b₂ -valued ManySortedSet of NAT
for s being State of S holds Comput P,s,(k + 1) = Exec (P . (IC (Comput P,s,k))),(Comput P,s,k)
let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty stored-program IC-Ins-separated definite steady-programmed AMI-Struct of N
for P being the Instructions of b₁ -valued ManySortedSet of NAT
for s being State of S holds Comput P,s,(k + 1) = Exec (P . (IC (Comput P,s,k))),(Comput P,s,k)
let S be non empty stored-program IC-Ins-separated definite steady-programmed AMI-Struct of N; :: thesis: for P being the Instructions of S -valued ManySortedSet of NAT
for s being State of S holds Comput P,s,(k + 1) = Exec (P . (IC (Comput P,s,k))),(Comput P,s,k)
let P be the Instructions of S -valued ManySortedSet of NAT ; :: thesis: for s being State of S holds Comput P,s,(k + 1) = Exec (P . (IC (Comput P,s,k))),(Comput P,s,k)
let s be State of S; :: thesis: Comput P,s,(k + 1) = Exec (P . (IC (Comput P,s,k))),(Comput P,s,k)
D: dom P = NAT by PARTFUN1:def 4;
IC (Comput P,s,k) in dom P by D;
then Y: P /. (IC (Comput P,s,k)) = P . (IC (Comput P,s,k)) by PARTFUN1:def 8;
thus Comput P,s,(k + 1) = Following P,(Comput P,s,k) by Th14
.= Exec (CurInstr P,(Comput P,s,k)),(Comput P,s,k)
.= Exec (P . (IC (Comput P,s,k))),(Comput P,s,k) by Y ; :: thesis: verum