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 stored-program IC-Ins-separated definite 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 stored-program IC-Ins-separated definite 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 13;
A2: j = i + k by A1;
let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty stored-program IC-Ins-separated definite 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 stored-program IC-Ins-separated definite 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 Th14
.= Comput p,s,i by A3, A5, Def8 ;
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