let s be State of SCM+FSA ; :: thesis: for I being Program of SCM+FSA st I is_closed_on s & I is_halting_on s holds
for m being Element of NAT st m < LifeSpan (s +* (I +* (Start-At (insloc 0 )))) holds
CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m)
let I be Program of SCM+FSA ; :: thesis: ( I is_closed_on s & I is_halting_on s implies for m being Element of NAT st m < LifeSpan (s +* (I +* (Start-At (insloc 0 )))) holds
CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m) )
set s1 = s +* (I +* (Start-At (insloc 0 )));
set s2 = s +* ((loop I) +* (Start-At (insloc 0 )));
assume A1: ( I is_closed_on s & I is_halting_on s ) ; :: thesis: for m being Element of NAT st m < LifeSpan (s +* (I +* (Start-At (insloc 0 )))) holds
CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m)
let m be Element of NAT ; :: thesis: ( m < LifeSpan (s +* (I +* (Start-At (insloc 0 )))) implies CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m) )
assume A2: m < LifeSpan (s +* (I +* (Start-At (insloc 0 )))) ; :: thesis: CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m)
( I +* (Start-At (insloc 0 )) c= s +* (I +* (Start-At (insloc 0 ))) & I c= I +* (Start-At (insloc 0 )) ) by FUNCT_4:26, SCMFSA8A:9;
then I c= s +* (I +* (Start-At (insloc 0 ))) by XBOOLE_1:1;
then A3: I c= Computation (s +* (I +* (Start-At (insloc 0 )))),m by AMI_1:81;
( (loop I) +* (Start-At (insloc 0 )) c= s +* ((loop I) +* (Start-At (insloc 0 ))) & loop I c= (loop I) +* (Start-At (insloc 0 )) ) by FUNCT_4:26, SCMFSA8A:9;
then loop I c= s +* ((loop I) +* (Start-At (insloc 0 ))) by XBOOLE_1:1;
then A4: loop I c= Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m by AMI_1:81;
A5: IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m) in dom I by A1, SCMFSA7B:def 7;
then A6: IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m) in dom (loop I) by FUNCT_4:105;
A7: s +* (I +* (Start-At (insloc 0 ))) is halting by A1, SCMFSA7B:def 8;
A8: CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = I . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m)) by A3, A5, GRFUNC_1:8;
then I . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m)) <> halt SCM+FSA by A2, A7, AMI_1:def 46;
then I . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m)) = (loop I) . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m)) by FUNCT_4:111;
hence CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m) = (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m) . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),m)) by A4, A6, A8, GRFUNC_1:8
.= CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m) by A1, A2, Th109, AMI_1:121 ;
:: thesis: verum