set Ma = Macro (SubFrom a,b);
let s be State of SCM+FSA ; :: according to SCMFSA6B:def 4,SCMFSA6C:def 2 :: thesis: ( not (Macro (SubFrom 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 A19: (Macro (SubFrom 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 A20: Macro (SubFrom 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 A21: IC SCM+FSA in dom (Start-At 0 ,SCM+FSA ) by TARSKI:def 1;
Start-At 0 ,SCM+FSA c= (Macro (SubFrom a,b)) +* (Start-At 0 ,SCM+FSA ) by FUNCT_4:26;
then Start-At 0 ,SCM+FSA c= s by A19, XBOOLE_1:1;
then A22: IC s = (Start-At 0 ,SCM+FSA ) . (IC SCM+FSA ) by A21, GRFUNC_1:8
.= 0 by FUNCOP_1:87 ;
0 in dom (Macro (SubFrom a,b)) by SCMFSA6B:32;
then A23: (Macro (SubFrom a,b)) . 0 = s . 0 by A20, GRFUNC_1:8;
Y: (ProgramPart s) /. (IC s) = s . (IC s) by AMI_1:150;
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 AMI_1:150;
A24: 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 (SubFrom a,b),s by A22, A23, SCMFSA6B:33, Y ;
1 in dom (Macro (SubFrom a,b)) by SCMFSA6B:32;
then (Macro (SubFrom a,b)) . 1 = s . 1 by A20, GRFUNC_1:8;
then A25: s . 1 = halt SCM+FSA by SCMFSA6B:33;
IC (Exec (SubFrom a,b),s) = succ 0 by A22, SCMFSA_2:91
.= 0 + 1 ;
then A26: CurInstr (ProgramPart (Comput (ProgramPart s),s,1)),(Comput (ProgramPart s),s,1) = halt SCM+FSA by A25, A24, AMI_1:def 13, Z;
T: ProgramPart s = ProgramPart (Comput (ProgramPart s),s,1) by AMI_1:144;
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 A27: 1 <= k ; :: thesis: (Comput (ProgramPart s),s,k) . (intloc 0 ) = s . (intloc 0 )
(Comput (ProgramPart s),s,1) . (intloc 0 ) = s . (intloc 0 ) by A24, SCMFSA_2:91;
hence (Comput (ProgramPart s),s,k) . (intloc 0 ) = s . (intloc 0 ) by A26, A27, AMI_1:52, T; :: thesis: verum
end;
end;