let i, j be Element of NAT ; :: thesis: for N being non empty with_non-empty_elements set st i <= j holds
for S being non empty stored-program halting IC-Ins-separated steady-programmed definite AMI-Struct of N
for s being State of S st s halts_at IC (Comput (ProgramPart s),s,i) holds
s halts_at IC (Comput (ProgramPart s),s,j)
let N be non empty with_non-empty_elements set ; :: thesis: ( i <= j implies for S being non empty stored-program halting IC-Ins-separated steady-programmed definite AMI-Struct of N
for s being State of S st s halts_at IC (Comput (ProgramPart s),s,i) holds
s halts_at IC (Comput (ProgramPart s),s,j) )
assume A1: i <= j ; :: 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 st s halts_at IC (Comput (ProgramPart s),s,i) holds
s halts_at IC (Comput (ProgramPart s),s,j)
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 st s halts_at IC (Comput (ProgramPart s),s,i) holds
s halts_at IC (Comput (ProgramPart s),s,j)
let s be State of S; :: thesis: ( s halts_at IC (Comput (ProgramPart s),s,i) implies s halts_at IC (Comput (ProgramPart s),s,j) )
Y: (ProgramPart (Comput (ProgramPart s),s,i)) /. (IC (Comput (ProgramPart s),s,i)) = (Comput (ProgramPart s),s,i) . (IC (Comput (ProgramPart s),s,i)) by BWL1;
X: ProgramPart (Comput (ProgramPart s),s,i) = ProgramPart s by LmY;
assume A2: s . (IC (Comput (ProgramPart s),s,i)) = halt S ; :: according to AMI_1:def 42 :: thesis: s halts_at IC (Comput (ProgramPart s),s,j)
then CurInstr (ProgramPart (Comput (ProgramPart s),s,i)),(Comput (ProgramPart s),s,i) = halt S by Th54, Y;
hence s . (IC (Comput (ProgramPart s),s,j)) = halt S by A1, A2, Th52, X; :: according to AMI_1:def 42 :: thesis: verum