set Ma = Macro (MultBy (a,b));
let s be State of SCM+FSA; :: according to SCMFSA6B:def 3,SCMFSA6C:def 1 :: thesis: ( not Start-At (0,SCM+FSA) c= s or for b₁ being set holds
( not Macro (MultBy (a,b)) c= b₁ or b₁ halts_on s ) )
assume A34: Start-At (0,SCM+FSA) c= s ; :: thesis: for b₁ being set holds
( not Macro (MultBy (a,b)) c= b₁ or b₁ halts_on s )
let P be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: ( not Macro (MultBy (a,b)) c= P or P halts_on s )
assume A35: Macro (MultBy (a,b)) c= P ; :: thesis: P halts_on s
A36: Macro (MultBy (a,b)) c= P by A35;
take 1 ; :: according to EXTPRO_1:def 7 :: thesis: ( IC (Comput (P,s,1)) in proj1 P & CurInstr (P,(Comput (P,s,1))) = halt SCM+FSA )
A37: dom P = NAT by PARTFUN1:def 4;
thus IC (Comput (P,s,1)) in dom P by A37; :: thesis: CurInstr (P,(Comput (P,s,1))) = halt SCM+FSA
dom (Start-At (0,SCM+FSA)) = {(IC )} by FUNCOP_1:19;
then A38: IC in dom (Start-At (0,SCM+FSA)) by TARSKI:def 1;
A39: IC s = (Start-At (0,SCM+FSA)) . (IC ) by A38, GRFUNC_1:8, A34
.= 0 by FUNCOP_1:87 ;
then A40: IC (Exec ((MultBy (a,b)),s)) = succ 0 by SCMFSA_2:92
.= 0 + 1 ;
A41: 1 in dom (Macro (MultBy (a,b))) by COMPOS_1:147;
A42: 0 in dom (Macro (MultBy (a,b))) by COMPOS_1:147;
A43: P . 0 = (Macro (MultBy (a,b))) . 0 by A36, A42, GRFUNC_1:8
.= MultBy (a,b) by COMPOS_1:148 ;
A44: P . 1 = (Macro (MultBy (a,b))) . 1 by A36, A41, GRFUNC_1:8
.= halt SCM+FSA by COMPOS_1:148 ;
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 A39, A37, PARTFUN1:def 8, A43 ;
hence CurInstr (P,(Comput (P,s,1))) = halt SCM+FSA by A40, A37, PARTFUN1:def 8, A44; :: thesis: verum