let i, j be Element of NAT ; :: thesis: ( i <= j implies for N being with_non-empty_elements set
for S being non empty stored-program halting IC-Ins-separated definite AMI-Struct of N
for s being State of S st CurInstr (Computation s,i) = halt S holds
Computation s,j = Computation s,i )
assume i <= j ; :: thesis: for N being with_non-empty_elements set
for S being non empty stored-program halting IC-Ins-separated definite AMI-Struct of N
for s being State of S st CurInstr (Computation s,i) = halt S holds
Computation s,j = Computation 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 with_non-empty_elements set ; :: thesis: for S being non empty stored-program halting IC-Ins-separated definite AMI-Struct of N
for s being State of S st CurInstr (Computation s,i) = halt S holds
Computation s,j = Computation s,i
let S be non empty stored-program halting IC-Ins-separated definite AMI-Struct of N; :: thesis: for s being State of S st CurInstr (Computation s,i) = halt S holds
Computation s,j = Computation s,i
let s be State of S; :: thesis: ( CurInstr (Computation s,i) = halt S implies Computation s,j = Computation s,i )
assume A3: CurInstr (Computation s,i) = halt S ; :: thesis: Computation s,j = Computation s,i
defpred S₁[ Element of NAT ] means Computation s,(i + $1) = Computation 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]
Computation s,(i + (k + 1)) = Computation s,((i + k) + 1)
.= Following (Computation s,(i + k)) by Th14
.= Computation 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 Computation s,j = Computation s,i by A2; :: thesis: verum