let i, j be Element of NAT ; :: thesis: ( i <= j implies for N being non empty with_non-empty_elements set
for S being non empty IC-Ins-separated halting AMI-Struct of N
for p being NAT -defined the Instructions of b₂ -valued Function
for s being State of S st CurInstr (p,(Comput (p,s,i))) = halt S holds
Comput (p,s,j) = Comput (p,s,i) )
assume i <= j ; :: thesis: for N being non empty with_non-empty_elements set
for S being non empty IC-Ins-separated halting AMI-Struct of N
for p being NAT -defined the Instructions of b₂ -valued Function
for s being State of S st CurInstr (p,(Comput (p,s,i))) = halt S holds
Comput (p,s,j) = Comput (p,s,i)
then consider k being Nat such that
A1: j = i + k by NAT_1:10;
reconsider k = k as Element of NAT by ORDINAL1:def 12;
A2: j = i + k by A1;
let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty IC-Ins-separated halting AMI-Struct of N
for p being NAT -defined the Instructions of b₁ -valued Function
for s being State of S st CurInstr (p,(Comput (p,s,i))) = halt S holds
Comput (p,s,j) = Comput (p,s,i)
let S be non empty IC-Ins-separated halting AMI-Struct of N; :: thesis: for p being NAT -defined the Instructions of S -valued Function
for s being State of S st CurInstr (p,(Comput (p,s,i))) = halt S holds
Comput (p,s,j) = Comput (p,s,i)
let p be NAT -defined the Instructions of S -valued Function; :: thesis: for s being State of S st CurInstr (p,(Comput (p,s,i))) = halt S holds
Comput (p,s,j) = Comput (p,s,i)
let s be State of S; :: thesis: ( CurInstr (p,(Comput (p,s,i))) = halt S implies Comput (p,s,j) = Comput (p,s,i) )
assume A3: CurInstr (p,(Comput (p,s,i))) = halt S ; :: thesis: Comput (p,s,j) = Comput (p,s,i)
defpred S₁[ Element of NAT ] means Comput (p,s,(i + $1)) = Comput (p,s,i);

A4: now

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A5: S₁[k] ; :: thesis: S₁[k + 1]
Comput (p,s,(i + (k + 1))) = Comput (p,s,((i + k) + 1))
.= Following (p,(Comput (p,s,(i + k)))) by Th4
.= Comput (p,s,i) by A3, A5, Def3 ;
hence S₁[k + 1] ; :: thesis: verum

end;

A6: S₁[ 0 ] ;
for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A6, A4);
hence Comput (p,s,j) = Comput (p,s,i) by A2; :: thesis: verum