let s be State of SCM+FSA ; :: according to SCMFSA6B:def 4,SCMFSA6C:def 2 :: thesis: ( not (Macro (AddTo a,b)) +* (Start-At (insloc 0 )) c= s or for b₁ being Element of NAT holds (Computation s,b₁) . (intloc 0 ) = s . (intloc 0 ) )
assume A11: (Macro (AddTo a,b)) +* (Start-At (insloc 0 )) c= s ; :: thesis: for b₁ being Element of NAT holds (Computation s,b₁) . (intloc 0 ) = s . (intloc 0 )
let k be Element of NAT ; :: thesis: (Computation s,k) . (intloc 0 ) = s . (intloc 0 )
set Ma = Macro (AddTo a,b);
Start-At (insloc 0 ) c= (Macro (AddTo a,b)) +* (Start-At (insloc 0 )) by FUNCT_4:26;
then A12: Start-At (insloc 0 ) c= s by A11, XBOOLE_1:1;
dom (Start-At (insloc 0 )) = {(IC SCM+FSA )} by FUNCOP_1:19;
then IC SCM+FSA in dom (Start-At (insloc 0 )) by TARSKI:def 1;
then A13: IC s = (Start-At (insloc 0 )) . (IC SCM+FSA ) by A12, GRFUNC_1:8
.= insloc 0 by FUNCOP_1:87 ;
A14: ( insloc 0 in dom (Macro (AddTo a,b)) & insloc 1 in dom (Macro (AddTo a,b)) ) by SCMFSA6B:32;
Macro (AddTo a,b) c= s by A11, SCMFSA6B:5;
then A15: ( (Macro (AddTo a,b)) . (insloc 0 ) = s . (insloc 0 ) & (Macro (AddTo a,b)) . (insloc 1) = s . (insloc 1) ) by A14, GRFUNC_1:8;
then A16: ( s . (insloc 0 ) = AddTo a,b & s . (insloc 1) = halt SCM+FSA ) by SCMFSA6B:33;
A17: IC (Exec (AddTo a,b),s) = Next (insloc 0 ) by A13, SCMFSA_2:90
.= insloc (0 + 1) ;
A18: Computation s,(0 + 1) = Following (Computation s,0 ) by AMI_1:14
.= Following s by AMI_1:13
.= Exec (AddTo a,b),s by A13, A15, SCMFSA6B:33 ;
then A19: CurInstr (Computation s,1) = halt SCM+FSA by A16, A17, AMI_1:def 13;

per cases ( k = 0 or 1 <= k ) by NAT_1:14;

suppose k = 0 ; :: thesis: (Computation s,k) . (intloc 0 ) = s . (intloc 0 )

hence (Computation s,k) . (intloc 0 ) = s . (intloc 0 ) by AMI_1:13; :: thesis: verum

end;

suppose A20: 1 <= k ; :: thesis: (Computation s,k) . (intloc 0 ) = s . (intloc 0 )

(Computation s,1) . (intloc 0 ) = s . (intloc 0 ) by A18, SCMFSA_2:90;
hence (Computation s,k) . (intloc 0 ) = s . (intloc 0 ) by A19, A20, AMI_1:52; :: thesis: verum

end;

end;