set Ma = Macro (MultBy a,b);
let s be State of SCM+FSA ; :: according to SCMFSA6B:def 4,SCMFSA6C:def 2 :: thesis: ( not (Macro (MultBy a,b)) +* (Start-At 0 ,SCM+FSA ) c= s or for b1 being Element of NAT holds (Comput (ProgramPart s),s,b1) . (intloc 0 ) = s . (intloc 0 ) )
assume A28: (Macro (MultBy a,b)) +* (Start-At 0 ,SCM+FSA ) c= s ; :: thesis: for b1 being Element of NAT holds (Comput (ProgramPart s),s,b1) . (intloc 0 ) = s . (intloc 0 )
then A29: Macro (MultBy a,b) c= s by SCMFSA6B:5;
let k be Element of NAT ; :: thesis: (Comput (ProgramPart s),s,k) . (intloc 0 ) = s . (intloc 0 )
dom (Start-At 0 ,SCM+FSA ) = {(IC SCM+FSA )} by FUNCOP_1:19;
then A30: IC SCM+FSA in dom (Start-At 0 ,SCM+FSA ) by TARSKI:def 1;
Start-At 0 ,SCM+FSA c= (Macro (MultBy a,b)) +* (Start-At 0 ,SCM+FSA ) by FUNCT_4:26;
then Start-At 0 ,SCM+FSA c= s by A28, XBOOLE_1:1;
then A31: IC s = (Start-At 0 ,SCM+FSA ) . (IC SCM+FSA ) by A30, GRFUNC_1:8
.= 0 by FUNCOP_1:87 ;
0 in dom (Macro (MultBy a,b)) by SCMFSA6B:32;
then A32: (Macro (MultBy a,b)) . 0 = s . 0 by A29, GRFUNC_1:8;
Y: (ProgramPart s) /. (IC s) = s . (IC s) by COMPOS_1:38;
Z: (ProgramPart (Comput (ProgramPart s),s,1)) /. (IC (Comput (ProgramPart s),s,1)) = (Comput (ProgramPart s),s,1) . (IC (Comput (ProgramPart s),s,1)) by COMPOS_1:38;
A33: Comput (ProgramPart s),s,(0 + 1) = Following (ProgramPart s),(Comput (ProgramPart s),s,0 ) by AMI_1:14
.= Following (ProgramPart s),s by AMI_1:13
.= Exec (MultBy a,b),s by A31, A32, Y, SCMFSA6B:33 ;
1 in dom (Macro (MultBy a,b)) by SCMFSA6B:32;
then (Macro (MultBy a,b)) . 1 = s . 1 by A29, GRFUNC_1:8;
then A34: s . 1 = halt SCM+FSA by SCMFSA6B:33;
IC (Exec (MultBy a,b),s) = succ 0 by A31, SCMFSA_2:92
.= 0 + 1 ;
then A35: CurInstr (ProgramPart (Comput (ProgramPart s),s,1)),(Comput (ProgramPart s),s,1) = halt SCM+FSA by A34, A33, Z, AMI_1:def 13;
T: ProgramPart s = ProgramPart (Comput (ProgramPart s),s,1) by AMI_1:123;
per cases ( k = 0 or 1 <= k ) by NAT_1:14;
suppose k = 0 ; :: thesis: (Comput (ProgramPart s),s,k) . (intloc 0 ) = s . (intloc 0 )
hence (Comput (ProgramPart s),s,k) . (intloc 0 ) = s . (intloc 0 ) by AMI_1:13; :: thesis: verum
end;
suppose A36: 1 <= k ; :: thesis: (Comput (ProgramPart s),s,k) . (intloc 0 ) = s . (intloc 0 )
(Comput (ProgramPart s),s,1) . (intloc 0 ) = s . (intloc 0 ) by A33, SCMFSA_2:92;
hence (Comput (ProgramPart s),s,k) . (intloc 0 ) = s . (intloc 0 ) by A35, A36, T, AMI_1:52; :: thesis: verum
end;
end;