let i, j be Nat; :: thesis: ( i <= j implies for N being non empty with_zero set
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 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_zero set
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 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 Nat ;
A2: j = i + k by A1;
let N be non empty with_zero set ; :: 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 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 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 CurInstr (p,(Comput (p,s,i))) = halt S 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 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[ Nat] means Comput (p,s,(i + \$1)) = Comput (p,s,i);
A4: now :: thesis: for k being Nat st S1[k] holds
S1[k + 1]
let k be 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 Th3
.= Comput (p,s,i) by A3, A5, Def3 ;
hence S1[k + 1] ; :: thesis: verum
end;
A6: S1[ 0 ] ;
for k being Nat holds S1[k] from NAT_1:sch 2(A6, A4);
hence Comput (p,s,j) = Comput (p,s,i) by A2; :: thesis: verum