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

P halts_at IC (Comput (P,s,j))

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

P halts_at IC (Comput (P,s,j)) )

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

P halts_at IC (Comput (P,s,j))

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

P halts_at IC (Comput (P,s,j))

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

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 that

A2: IC (Comput (p,s,i)) in dom p and

A3: p . (IC (Comput (p,s,i))) = halt S ; :: according to COMPOS_1:def 12 :: thesis: p halts_at IC (Comput (p,s,j))

A4: CurInstr (p,(Comput (p,s,i))) = halt S by A2, A3, PARTFUN1:def 6;

hence IC (Comput (p,s,j)) in dom p by A2, A1, Th5; :: according to COMPOS_1:def 12 :: thesis: p . (IC (Comput (p,s,j))) = halt S

thus p . (IC (Comput (p,s,j))) = halt S by A1, A3, A4, Th5; :: 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

P halts_at IC (Comput (P,s,j))

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

P halts_at IC (Comput (P,s,j)) )

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

P halts_at IC (Comput (P,s,j))

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

P halts_at IC (Comput (P,s,j))

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

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 that

A2: IC (Comput (p,s,i)) in dom p and

A3: p . (IC (Comput (p,s,i))) = halt S ; :: according to COMPOS_1:def 12 :: thesis: p halts_at IC (Comput (p,s,j))

A4: CurInstr (p,(Comput (p,s,i))) = halt S by A2, A3, PARTFUN1:def 6;

hence IC (Comput (p,s,j)) in dom p by A2, A1, Th5; :: according to COMPOS_1:def 12 :: thesis: p . (IC (Comput (p,s,j))) = halt S

thus p . (IC (Comput (p,s,j))) = halt S by A1, A3, A4, Th5; :: thesis: verum