let s be State of SCM+FSA ; :: according to SCMFSA6B:def 4,SCMFSA6C:def 2 :: thesis: ( not (Macro (b := f,a)) +* (Start-At (insloc 0 )) c= s or for b₁ being Element of NAT holds (Computation s,b₁) . (intloc 0 ) = s . (intloc 0 ) )
assume A1: (Macro (b := f,a)) +* (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 (b := f,a);
Start-At (insloc 0 ) c= (Macro (b := f,a)) +* (Start-At (insloc 0 )) by FUNCT_4:26;
then A2: Start-At (insloc 0 ) c= s by A1, 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 A3: IC s = (Start-At (insloc 0 )) . (IC SCM+FSA ) by A2, GRFUNC_1:8
.= insloc 0 by FUNCOP_1:87 ;
A4: ( insloc 0 in dom (Macro (b := f,a)) & insloc 1 in dom (Macro (b := f,a)) ) by SCMFSA6B:32;
Macro (b := f,a) c= s by A1, SCMFSA6B:5;
then A5: ( (Macro (b := f,a)) . (insloc 0 ) = s . (insloc 0 ) & (Macro (b := f,a)) . (insloc 1) = s . (insloc 1) ) by A4, GRFUNC_1:8;
then A6: ( s . (insloc 0 ) = b := f,a & s . (insloc 1) = halt SCM+FSA ) by SCMFSA6B:33;
A7: IC (Exec (b := f,a),s) = Next (insloc 0 ) by A3, SCMFSA_2:98
.= insloc (0 + 1) ;
A8: Computation s,(0 + 1) = Following (Computation s,0 ) by AMI_1:14
.= Following s by AMI_1:13
.= Exec (b := f,a),s by A3, A5, SCMFSA6B:33 ;
then A9: CurInstr (Computation s,1) = halt SCM+FSA by A6, A7, 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 A10: 1 <= k ; :: thesis: (Computation s,k) . (intloc 0 ) = s . (intloc 0 )

(Computation s,1) . (intloc 0 ) = s . (intloc 0 ) by A8, SCMFSA_2:98;
hence (Computation s,k) . (intloc 0 ) = s . (intloc 0 ) by A9, A10, AMI_1:52; :: thesis: verum

end;

end;