let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty stored-program halting IC-Ins-separated definite AMI-Struct of N
for p being NAT -defined the Instructions of b₁ -valued finite Function
for s being State of S st ex k being Element of NAT st
( IC (Comput p,s,k) in dom p & p . (IC (Comput p,s,k)) = halt S ) holds
for i being Nat holds Result p,s = Result p,(Comput p,s,i)
let S be non empty stored-program halting IC-Ins-separated definite AMI-Struct of N; :: thesis: for p being NAT -defined the Instructions of S -valued finite Function
for s being State of S st ex k being Element of NAT st
( IC (Comput p,s,k) in dom p & p . (IC (Comput p,s,k)) = halt S ) holds
for i being Nat holds Result p,s = Result p,(Comput p,s,i)
let p be NAT -defined the Instructions of S -valued finite Function; :: thesis: for s being State of S st ex k being Element of NAT st
( IC (Comput p,s,k) in dom p & p . (IC (Comput p,s,k)) = halt S ) holds
for i being Nat holds Result p,s = Result p,(Comput p,s,i)
let s be State of S; :: thesis: ( ex k being Element of NAT st
( IC (Comput p,s,k) in dom p & p . (IC (Comput p,s,k)) = halt S ) implies for i being Nat holds Result p,s = Result p,(Comput p,s,i) )
given k being Element of NAT such that A0: IC (Comput p,s,k) in dom p and
A1: p . (IC (Comput p,s,k)) = halt S ; :: thesis: for i being Nat holds Result p,s = Result p,(Comput p,s,i)
let i be Nat; :: thesis: Result p,s = Result p,(Comput p,s,i)
set s9 = Comput p,s,k;
A2: CurInstr p,(Comput p,s,k) = halt S by A0, A1, PARTFUN1:def 8;

now

per cases ( i <= k or i >= k ) ;

suppose i <= k ; :: thesis: Result p,s = Result p,(Comput p,s,i)

then consider j being Nat such that
A3: k = i + j by NAT_1:10;
reconsider j = j as Element of NAT by ORDINAL1:def 13;
A4: Comput p,s,k = Comput p,(Comput p,s,i),j by A3, Th51;
then A5: p halts_on Comput p,s,i by A0, A2, Def8;
thus Result p,s = Comput p,s,k by A0, A1, Th56
.= Result p,(Comput p,s,i) by A2, A4, A5, Def10 ; :: thesis: verum

end;

suppose A6: i >= k ; :: thesis: Result p,s = Result p,(Comput p,s,i)

A7: Comput p,(Comput p,s,k),0 = Comput p,s,k by AMI_1:13;
A8: Comput p,s,i = Comput p,s,k by A2, A6, Th7;
then A9: p halts_on Comput p,s,i by A0, A2, A7, Def8;
thus Result p,s = Comput p,s,k by A0, A1, Th56
.= Result p,(Comput p,s,i) by A2, A8, A7, A9, Def10 ; :: thesis: verum

end;

end;

end;

hence Result p,s = Result p,(Comput p,s,i) ; :: thesis: verum