let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty stored-program halting IC-Ins-separated steady-programmed definite AMI-Struct of N
for s being State of S
for m being Element of NAT holds
( ProgramPart s halts_on s iff ProgramPart s halts_on Computation s,m )
let S be non empty stored-program halting IC-Ins-separated steady-programmed definite AMI-Struct of N; :: thesis: for s being State of S
for m being Element of NAT holds
( ProgramPart s halts_on s iff ProgramPart s halts_on Computation s,m )
let s be State of S; :: thesis: for m being Element of NAT holds
( ProgramPart s halts_on s iff ProgramPart s halts_on Computation s,m )
let m be Element of NAT ; :: thesis: ( ProgramPart s halts_on s iff ProgramPart s halts_on Computation s,m )

hereby :: thesis: ( ProgramPart s halts_on Computation s,m implies ProgramPart s halts_on s )

assume ProgramPart s halts_on s ; :: thesis: ProgramPart s halts_on Computation s,m
then consider n being Element of NAT such that
W0: IC (Computation s,n) in dom (ProgramPart s) and
W: (ProgramPart s) . (IC (Computation s,n)) = halt S by Def20;
A1: CurInstr (Computation s,n) = halt S by W, LmX;

per cases ( n <= m or n >= m ) ;

suppose n <= m ; :: thesis: ProgramPart s halts_on Computation s,m

then A: Computation s,n = Computation s,(m + 0 ) by A1, Th52
.= Computation (Computation s,m),0 by Th51 ;
ProgramPart (Computation s,m) = ProgramPart s by LmY;
hence ProgramPart s halts_on Computation s,m by Def20, W, W0, A; :: thesis: verum

end;

suppose n >= m ; :: thesis: ProgramPart s halts_on Computation s,m

then reconsider k = n - m as Element of NAT by INT_1:18;
A: Computation (Computation s,m),k = Computation s,(m + k) by Th51
.= Computation s,n ;
ProgramPart (Computation s,m) = ProgramPart s by LmY;
hence ProgramPart s halts_on Computation s,m by Def20, W0, W, A; :: thesis: verum

end;

end;

end;

given n being Element of NAT such that W1: IC (Computation (Computation s,m),n) in dom (ProgramPart s) and
W2: (ProgramPart s) . (IC (Computation (Computation s,m),n)) = halt S ; :: according to AMI_1:def 20 :: thesis: ProgramPart s halts_on s
take m + n ; :: according to AMI_1:def 20 :: thesis: ( IC (Computation s,(m + n)) in dom (ProgramPart s) & (ProgramPart s) . (IC (Computation s,(m + n))) = halt S )
ProgramPart (Computation s,m) = ProgramPart s by LmY;
hence ( IC (Computation s,(m + n)) in dom (ProgramPart s) & (ProgramPart s) . (IC (Computation s,(m + n))) = halt S ) by W1, W2, Th51; :: thesis: verum