let N be 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 k being Element of NAT st ProgramPart s halts_on s holds
( Result s = Computation s,k iff s halts_at IC (Computation s,k) )
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 k being Element of NAT st ProgramPart s halts_on s holds
( Result s = Computation s,k iff s halts_at IC (Computation s,k) )
let s be State of S; :: thesis: for k being Element of NAT st ProgramPart s halts_on s holds
( Result s = Computation s,k iff s halts_at IC (Computation s,k) )
let k be Element of NAT ; :: thesis: ( ProgramPart s halts_on s implies ( Result s = Computation s,k iff s halts_at IC (Computation s,k) ) )
assume A1: ProgramPart s halts_on s ; :: thesis: ( Result s = Computation s,k iff s halts_at IC (Computation s,k) )

hereby :: thesis: ( s halts_at IC (Computation s,k) implies Result s = Computation s,k )

assume Result s = Computation s,k ; :: thesis: s halts_at IC (Computation s,k)
then ex i being Element of NAT st
( Computation s,k = Computation s,i & CurInstr (Computation s,k) = halt S ) by A1, Def22;
then s . (IC (Computation s,k)) = halt S by Th54;
hence s halts_at IC (Computation s,k) by Def42; :: thesis: verum

end;

assume s . (IC (Computation s,k)) = halt S ; :: according to AMI_1:def 42 :: thesis: Result s = Computation s,k
then CurInstr (Computation s,k) = halt S by Th54;
hence Result s = Computation s,k by A1, Def22; :: thesis: verum