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 A1: 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 A2: 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 AMI_1:def 26 :: thesis: ( not Relocated p,k c= t or t is halting )
assume A3: Relocated p,k c= t ; :: thesis: t is halting
reconsider s = t +* p as State of SCM+FSA ;
A4: p c= t +* p by FUNCT_4:26;
then s is halting by A2, AMI_1:def 26;
then consider u being Element of NAT such that
A5: CurInstr (Computation s,u) = halt SCM+FSA by AMI_1:def 20;
reconsider s3 = s +* (DataPart t) as State of SCM+FSA ;
A6: CurInstr (Computation t,u) = IncAddr (halt SCM+FSA ),k by A1, A3, A4, A5, Th12
.= halt SCM+FSA by SCMFSA_4:8 ;
take u ; :: according to AMI_1:def 20 :: thesis: CurInstr (Computation t,u) = halt SCM+FSA
thus CurInstr (Computation t,u) = halt SCM+FSA by A6; :: thesis: verum

end;

end;

assume A7: Relocated p,k is halting ; :: thesis: p is halting
let t be State of SCM+FSA ; :: according to AMI_1:def 26 :: thesis: ( not p c= t or t is halting )
assume A8: p c= t ; :: thesis: t is halting
reconsider s = t +* (Relocated p,k) as State of SCM+FSA ;
A9: Relocated p,k c= t +* (Relocated p,k) by FUNCT_4:26;
then s is halting by A7, AMI_1:def 26;
then consider u being Element of NAT such that
A10: CurInstr (Computation s,u) = halt SCM+FSA by AMI_1:def 20;
reconsider s3 = t +* (DataPart s) as State of SCM+FSA ;
A11: IncAddr (CurInstr (Computation t,u)),k = halt SCM+FSA by A1, A8, A9, A10, Th12;
take u ; :: according to AMI_1:def 20 :: thesis: CurInstr (Computation t,u) = halt SCM+FSA
A12: not 0 in {6,7,8} ;
InsCode (CurInstr (Computation t,u)) = 0 by A11, SCMFSA_2:124, SCMFSA_4:22;
hence CurInstr (Computation t,u) = halt SCM+FSA by A11, A12, SCMFSA_4:def 3; :: thesis: verum