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 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;
A24: Comput ((ProgramPart s),s,(0 + 1)) = Following ((ProgramPart s),(Comput ((ProgramPart s),s,0))) by EXTPRO_1:4
.= Following ((ProgramPart s),s) by EXTPRO_1:3
.= Exec ((SubFrom (a,b)),s) by A22, A23, Y, COMPOS_1:148, SCMFSA6B:33 ;
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 COMPOS_1:148, 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, 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 EXTPRO_1:3; :: 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, T, EXTPRO_1:6; :: thesis: verum
end;
end;