let i, j be Nat; :: thesis: for N being non empty with_zero set st i <= j holds

for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N

for P being NAT -defined the InstructionsF of b_{2} -valued Function

for s being State of S st P halts_at IC (Comput (P,s,i)) holds

Comput (P,s,j) = Comput (P,s,i)

let N be non empty with_zero set ; :: thesis: ( i <= j implies for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N

for P being NAT -defined the InstructionsF of b_{1} -valued Function

for s being State of S st P halts_at IC (Comput (P,s,i)) holds

Comput (P,s,j) = Comput (P,s,i) )

assume A1: i <= j ; :: thesis: for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N

for P being NAT -defined the InstructionsF of b_{1} -valued Function

for s being State of S st P halts_at IC (Comput (P,s,i)) holds

Comput (P,s,j) = Comput (P,s,i)

let S be non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N; :: thesis: for P being NAT -defined the InstructionsF of S -valued Function

for s being State of S st P halts_at IC (Comput (P,s,i)) holds

Comput (P,s,j) = Comput (P,s,i)

let P be NAT -defined the InstructionsF of S -valued Function; :: thesis: for s being State of S st P halts_at IC (Comput (P,s,i)) holds

Comput (P,s,j) = Comput (P,s,i)

let s be State of S; :: thesis: ( P halts_at IC (Comput (P,s,i)) implies Comput (P,s,j) = Comput (P,s,i) )

assume A2: P halts_at IC (Comput (P,s,i)) ; :: thesis: Comput (P,s,j) = Comput (P,s,i)

then P halts_at IC (Comput (P,s,j)) by A1, Th19;

hence Comput (P,s,j) = Result (P,s) by Th18

.= Comput (P,s,i) by A2, Th18 ;

:: thesis: verum

for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N

for P being NAT -defined the InstructionsF of b

for s being State of S st P halts_at IC (Comput (P,s,i)) holds

Comput (P,s,j) = Comput (P,s,i)

let N be non empty with_zero set ; :: thesis: ( i <= j implies for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N

for P being NAT -defined the InstructionsF of b

for s being State of S st P halts_at IC (Comput (P,s,i)) holds

Comput (P,s,j) = Comput (P,s,i) )

assume A1: i <= j ; :: thesis: for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N

for P being NAT -defined the InstructionsF of b

for s being State of S st P halts_at IC (Comput (P,s,i)) holds

Comput (P,s,j) = Comput (P,s,i)

let S be non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N; :: thesis: for P being NAT -defined the InstructionsF of S -valued Function

for s being State of S st P halts_at IC (Comput (P,s,i)) holds

Comput (P,s,j) = Comput (P,s,i)

let P be NAT -defined the InstructionsF of S -valued Function; :: thesis: for s being State of S st P halts_at IC (Comput (P,s,i)) holds

Comput (P,s,j) = Comput (P,s,i)

let s be State of S; :: thesis: ( P halts_at IC (Comput (P,s,i)) implies Comput (P,s,j) = Comput (P,s,i) )

assume A2: P halts_at IC (Comput (P,s,i)) ; :: thesis: Comput (P,s,j) = Comput (P,s,i)

then P halts_at IC (Comput (P,s,j)) by A1, Th19;

hence Comput (P,s,j) = Result (P,s) by Th18

.= Comput (P,s,i) by A2, Th18 ;

:: thesis: verum