let i, j be Element of NAT ; :: thesis: for N being non empty with_non-empty_elements set st i <= j holds
for S being non empty stored-program halting IC-Ins-separated steady-programmed definite AMI-Struct of N
for P being the Instructions of b₂ -valued ManySortedSet of NAT
for s being State of S st P halts_at IC (Comput P,s,i) holds
P halts_at IC (Comput P,s,j)
let N be non empty with_non-empty_elements set ; :: thesis: ( i <= j implies for S being non empty stored-program halting IC-Ins-separated steady-programmed definite AMI-Struct of N
for P being the Instructions of b₁ -valued ManySortedSet of NAT
for s being State of S st P halts_at IC (Comput P,s,i) holds
P halts_at IC (Comput P,s,j) )
assume A1: i <= j ; :: thesis: for S being non empty stored-program halting IC-Ins-separated steady-programmed definite AMI-Struct of N
for P being the Instructions of b₁ -valued ManySortedSet of NAT
for s being State of S st P halts_at IC (Comput P,s,i) holds
P halts_at IC (Comput P,s,j)
let S be non empty stored-program halting IC-Ins-separated steady-programmed definite AMI-Struct of N; :: thesis: for P being the Instructions of S -valued ManySortedSet of NAT
for s being State of S st P halts_at IC (Comput P,s,i) holds
P halts_at IC (Comput P,s,j)
let P be the Instructions of S -valued ManySortedSet of NAT ; :: thesis: for s being State of S st P halts_at IC (Comput P,s,i) holds
P halts_at IC (Comput P,s,j)
let s be State of S; :: thesis: ( P halts_at IC (Comput P,s,i) implies P halts_at IC (Comput P,s,j) )
assume Z: P . (IC (Comput P,s,i)) = halt S ; :: according to AMI_1:def 45 :: thesis: P halts_at IC (Comput P,s,j)
dom P = NAT by PARTFUN1:def 4;
then IC (Comput P,s,i) in dom P ;
then P /. (IC (Comput P,s,i)) = halt S by Z, PARTFUN1:def 8;
then CurInstr P,(Comput P,s,i) = halt S ;
hence P . (IC (Comput P,s,j)) = halt S by A1, Th52, Z; :: according to AMI_1:def 45 :: thesis: verum