let p be autonomic FinPartState of SCM+FSA; :: thesis: for k being Element of NAT st IC SCM+FSA in dom p holds
( p is halting iff Relocated (p,k) is halting )
let k be Element of NAT ; :: thesis: ( IC SCM+FSA in dom p implies ( p is halting iff Relocated (p,k) is halting ) )
assume A2: IC SCM+FSA in dom p ; :: thesis: ( p is halting iff Relocated (p,k) is halting )

hereby :: thesis: ( Relocated (p,k) is halting implies p is halting )

assume A3: p is halting ; :: thesis: Relocated (p,k) is halting
thus Relocated (p,k) is halting :: thesis: verum

proof

let t be State of SCM+FSA; :: according to EXTPRO_1:def 10 :: thesis: ( not Relocated (p,k) c= t or ProgramPart t halts_on t )
assume A4: Relocated (p,k) c= t ; :: thesis: ProgramPart t halts_on t
reconsider s = t +* p as State of SCM+FSA ;
A5: p c= t +* p by FUNCT_4:26;
then ProgramPart s halts_on s by A3, EXTPRO_1:def 10;
then consider u being Element of NAT such that
A6: CurInstr ((ProgramPart s),(Comput ((ProgramPart s),s,u))) = halt SCM+FSA by EXTPRO_1:30;
take u ; :: according to EXTPRO_1:def 7 :: thesis: ( IC (Comput ((ProgramPart t),t,u)) in proj1 (ProgramPart t) & CurInstr ((ProgramPart t),(Comput ((ProgramPart t),t,u))) = halt SCM+FSA )
IC (Comput ((ProgramPart t),t,u)) in NAT ;
hence IC (Comput ((ProgramPart t),t,u)) in dom (ProgramPart t) by COMPOS_1:34; :: thesis: CurInstr ((ProgramPart t),(Comput ((ProgramPart t),t,u))) = halt SCM+FSA
CurInstr ((ProgramPart t),(Comput ((ProgramPart t),t,u))) = IncAddr ((halt SCM+FSA),k) by A2, A4, A5, A6, Th12
.= halt SCM+FSA by SCMFSA_4:8 ;
hence CurInstr ((ProgramPart t),(Comput ((ProgramPart t),t,u))) = halt SCM+FSA ; :: thesis: verum
reconsider s3 = s +* (DataPart t) as State of SCM+FSA ;

end;

end;

assume A7: Relocated (p,k) is halting ; :: thesis: p is halting
let t be State of SCM+FSA; :: according to EXTPRO_1:def 10 :: thesis: ( not p c= t or ProgramPart t halts_on t )
reconsider s = t +* (Relocated (p,k)) as State of SCM+FSA ;
A8: Relocated (p,k) c= t +* (Relocated (p,k)) by FUNCT_4:26;
then ProgramPart s halts_on s by A7, EXTPRO_1:def 10;
then consider u being Element of NAT such that
A9: CurInstr ((ProgramPart s),(Comput ((ProgramPart s),s,u))) = halt SCM+FSA by EXTPRO_1:30;
assume p c= t ; :: thesis: ProgramPart t halts_on t
then IncAddr ((CurInstr ((ProgramPart t),(Comput ((ProgramPart t),t,u)))),k) = halt SCM+FSA by A2, A8, A9, Th12;
then B10: IncAddr ((CurInstr ((ProgramPart t),(Comput ((ProgramPart t),t,u)))),k) = IncAddr ((halt SCM+FSA),k) by COMPOS_1:93;
take u ; :: according to EXTPRO_1:def 7 :: thesis: ( IC (Comput ((ProgramPart t),t,u)) in proj1 (ProgramPart t) & CurInstr ((ProgramPart t),(Comput ((ProgramPart t),t,u))) = halt SCM+FSA )
IC (Comput ((ProgramPart t),t,u)) in NAT ;
hence IC (Comput ((ProgramPart t),t,u)) in dom (ProgramPart t) by COMPOS_1:34; :: thesis: CurInstr ((ProgramPart t),(Comput ((ProgramPart t),t,u))) = halt SCM+FSA
thus CurInstr ((ProgramPart t),(Comput ((ProgramPart t),t,u))) = halt SCM+FSA by B10, COMPOS_1:95; :: thesis: verum