set Ma = Macro (MultBy (a,b));
let s be State of SCM+FSA; :: according to SCMFSA6B:def 4,SCMFSA6C:def 2 :: thesis: ( not Start-At (0,SCM+FSA) c= s or for b₁ being set holds
( not Macro (MultBy (a,b)) c= b₁ or for b₂ being Element of NAT holds (Comput (b₁,s,b₂)) . (intloc 0) = s . (intloc 0) ) )
assume A31: Start-At (0,SCM+FSA) c= s ; :: thesis: for b₁ being set holds
( not Macro (MultBy (a,b)) c= b₁ or for b₂ being Element of NAT holds (Comput (b₁,s,b₂)) . (intloc 0) = s . (intloc 0) )
let P be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: ( not Macro (MultBy (a,b)) c= P or for b₁ being Element of NAT holds (Comput (P,s,b₁)) . (intloc 0) = s . (intloc 0) )
assume A32: Macro (MultBy (a,b)) c= P ; :: thesis: for b₁ being Element of NAT holds (Comput (P,s,b₁)) . (intloc 0) = s . (intloc 0)
let k be Element of NAT ; :: thesis: (Comput (P,s,k)) . (intloc 0) = s . (intloc 0)
dom (Start-At (0,SCM+FSA)) = {(IC )} by FUNCOP_1:19;
then A33: IC in dom (Start-At (0,SCM+FSA)) by TARSKI:def 1;
A34: IC s = (Start-At (0,SCM+FSA)) . (IC ) by A33, GRFUNC_1:8, A31
.= 0 by FUNCOP_1:87 ;
0 in dom (Macro (MultBy (a,b))) by COMPOS_1:147;
then A35: (Macro (MultBy (a,b))) . 0 = P . 0 by A32, GRFUNC_1:8;
A36: P /. (IC s) = P . (IC s) by PBOOLE:158;
A37: Comput (P,s,(0 + 1)) = Following (P,(Comput (P,s,0))) by EXTPRO_1:4
.= Following (P,s) by EXTPRO_1:3
.= Exec ((MultBy (a,b)),s) by A34, A35, A36, COMPOS_1:148 ;
1 in dom (Macro (MultBy (a,b))) by COMPOS_1:147;
then (Macro (MultBy (a,b))) . 1 = P . 1 by A32, GRFUNC_1:8;
then A38: P . 1 = halt SCM+FSA by COMPOS_1:148;
IC (Exec ((MultBy (a,b)),s)) = succ 0 by A34, SCMFSA_2:92
.= 0 + 1 ;
then A39: CurInstr (P,(Comput (P,s,1))) = halt SCM+FSA by A38, A37, PBOOLE:158;

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

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

hence (Comput (P,s,k)) . (intloc 0) = s . (intloc 0) by EXTPRO_1:3; :: thesis: verum

end;

suppose A40: 1 <= k ; :: thesis: (Comput (P,s,k)) . (intloc 0) = s . (intloc 0)

(Comput (P,s,1)) . (intloc 0) = s . (intloc 0) by A37, SCMFSA_2:92;
hence (Comput (P,s,k)) . (intloc 0) = s . (intloc 0) by A39, A40, EXTPRO_1:6; :: thesis: verum

end;

end;