let s be State of SCM+FSA ; :: according to SCMFSA6B:def 4 :: thesis: ( (I ';' J) +* (Start-At 0 ,SCM+FSA ) c= s implies for k being Element of NAT holds (Comput (ProgramPart s),s,k) . (intloc 0 ) = s . (intloc 0 ) )
set SA0 = Start-At 0 ,SCM+FSA ;
A1: I +* (Start-At 0 ,SCM+FSA ) c= s +* (I +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:26;
assume A2: (I ';' J) +* (Start-At 0 ,SCM+FSA ) c= s ; :: thesis: for k being Element of NAT holds (Comput (ProgramPart s),s,k) . (intloc 0 ) = s . (intloc 0 )
then A3: s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )) = s by FUNCT_4:79;
( dom (Start-At 0 ,SCM+FSA ) = {(IC SCM+FSA )} & intloc 0 <> IC SCM+FSA ) by FUNCOP_1:19, SCMFSA_2:81;
then ( not intloc 0 in dom I & not intloc 0 in dom (Start-At 0 ,SCM+FSA ) ) by SCMFSA6A:47, TARSKI:def 1;
then not intloc 0 in (dom I) \/ (dom (Start-At 0 ,SCM+FSA )) by XBOOLE_0:def 3;
then A4: not intloc 0 in dom (I +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:def 1;
Start-At 0 ,SCM+FSA c= (I ';' J) +* (Start-At 0 ,SCM+FSA ) by FUNCT_4:26;
then A5: Start-At 0 ,SCM+FSA c= s by A2, XBOOLE_1:1;
A6: s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )) = (s +* (I ';' J)) +* (Start-At 0 ,SCM+FSA ) by FUNCT_4:15
.= (s +* (Start-At 0 ,SCM+FSA )) +* (I ';' J) by Th14
.= s +* (I ';' J) by A5, FUNCT_4:79 ;

per cases ( ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA ))) halts_on s +* (I +* (Start-At 0 ,SCM+FSA )) or not ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA ))) halts_on s +* (I +* (Start-At 0 ,SCM+FSA )) ) ;

suppose A7: ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA ))) halts_on s +* (I +* (Start-At 0 ,SCM+FSA )) ; :: thesis: for k being Element of NAT holds (Comput (ProgramPart s),s,k) . (intloc 0 ) = s . (intloc 0 )

let k be Element of NAT ; :: thesis: (Comput (ProgramPart s),s,k) . (intloc 0 ) = s . (intloc 0 )
A8: s +* (I +* (Start-At 0 ,SCM+FSA )) = (s +* I) +* (Start-At 0 ,SCM+FSA ) by FUNCT_4:15
.= (s +* (Start-At 0 ,SCM+FSA )) +* I by Th14
.= s +* I by A5, FUNCT_4:79 ;

hereby :: thesis: verum

per cases ( k <= LifeSpan (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))) or k > LifeSpan (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))) ) ;

suppose A9: k <= LifeSpan (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))) ; :: thesis: (Comput (ProgramPart s),s,k) . (intloc 0 ) = s . (intloc 0 )

(Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k) . (intloc 0 ) = (s +* (I +* (Start-At 0 ,SCM+FSA ))) . (intloc 0 ) by A1, Def4
.= s . (intloc 0 ) by A4, FUNCT_4:12 ;
hence (Comput (ProgramPart s),s,k) . (intloc 0 ) = s . (intloc 0 ) by A3, A7, A9, Th40, SCMFSA10:92; :: thesis: verum

end;

suppose A10: k > LifeSpan (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))) ; :: thesis: (Comput (ProgramPart s),s,k) . (intloc 0 ) = s . (intloc 0 )

set LS = LifeSpan (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA )));
consider p being Element of NAT such that
A11: k = (LifeSpan (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA )))) + p and
A12: 1 <= p by A10, FINSEQ_4:99;
consider r being Nat such that
A13: p = 1 + r by A12, NAT_1:10;
( dom (Start-At 0 ,SCM+FSA ) = {(IC SCM+FSA )} & intloc 0 <> IC SCM+FSA ) by FUNCOP_1:19, SCMFSA_2:81;
then ( not intloc 0 in dom J & not intloc 0 in dom (Start-At 0 ,SCM+FSA ) ) by SCMFSA6A:47, TARSKI:def 1;
then not intloc 0 in (dom J) \/ (dom (Start-At 0 ,SCM+FSA )) by XBOOLE_0:def 3;
then A14: not intloc 0 in dom (J +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:def 1;
reconsider r = r as Element of NAT by ORDINAL1:def 13;
( dom (Start-At ((IC (Comput (ProgramPart ((Result (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA )))) +* (J +* (Start-At 0 ,SCM+FSA )))),((Result (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA )))) +* (J +* (Start-At 0 ,SCM+FSA ))),r)) + (card I)),SCM+FSA ) = {(IC SCM+FSA )} & intloc 0 <> IC SCM+FSA ) by FUNCOP_1:19, SCMFSA_2:81;
then not intloc 0 in dom (Start-At ((IC (Comput (ProgramPart ((Result (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA )))) +* (J +* (Start-At 0 ,SCM+FSA )))),((Result (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA )))) +* (J +* (Start-At 0 ,SCM+FSA ))),r)) + (card I)),SCM+FSA ) by TARSKI:def 1;
then A15: ((Comput (ProgramPart ((Result (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA )))) +* (J +* (Start-At 0 ,SCM+FSA )))),((Result (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA )))) +* (J +* (Start-At 0 ,SCM+FSA ))),r) +* (Start-At ((IC (Comput (ProgramPart ((Result (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA )))) +* (J +* (Start-At 0 ,SCM+FSA )))),((Result (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA )))) +* (J +* (Start-At 0 ,SCM+FSA ))),r)) + (card I)),SCM+FSA )) . (intloc 0 ) = (Comput (ProgramPart ((Result (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA )))) +* (J +* (Start-At 0 ,SCM+FSA )))),((Result (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA )))) +* (J +* (Start-At 0 ,SCM+FSA ))),r) . (intloc 0 ) by FUNCT_4:12;
A16: (Comput (ProgramPart ((Result (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA )))) +* (J +* (Start-At 0 ,SCM+FSA )))),((Result (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA )))) +* (J +* (Start-At 0 ,SCM+FSA ))),r) +* (Start-At ((IC (Comput (ProgramPart ((Result (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA )))) +* (J +* (Start-At 0 ,SCM+FSA )))),((Result (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA )))) +* (J +* (Start-At 0 ,SCM+FSA ))),r)) + (card I)),SCM+FSA ), Comput (ProgramPart (s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA )))),(s +* ((I ';' J) +* (Start-At 0 ,SCM+FSA ))),(((LifeSpan (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA )))) + 1) + r) equal_outside NAT by A2, A6, A7, A8, Th42;
J +* (Start-At 0 ,SCM+FSA ) c= (Result (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA )))) +* (J +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:26;
then (Comput (ProgramPart ((Result (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA )))) +* (J +* (Start-At 0 ,SCM+FSA )))),((Result (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA )))) +* (J +* (Start-At 0 ,SCM+FSA ))),r) . (intloc 0 ) = ((Result (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA )))) +* (J +* (Start-At 0 ,SCM+FSA ))) . (intloc 0 ) by Def4
.= (Result (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA )))) . (intloc 0 ) by A14, FUNCT_4:12
.= (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(LifeSpan (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))))) . (intloc 0 ) by A7, AMI_1:122
.= (s +* (I +* (Start-At 0 ,SCM+FSA ))) . (intloc 0 ) by A1, Def4
.= s . (intloc 0 ) by A4, FUNCT_4:12 ;
hence (Comput (ProgramPart s),s,k) . (intloc 0 ) = s . (intloc 0 ) by A3, A11, A13, A15, A16, SCMFSA10:92; :: thesis: verum

end;

end;

end;

end;

suppose A17: not ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA ))) halts_on s +* (I +* (Start-At 0 ,SCM+FSA )) ; :: thesis: for k being Element of NAT holds (Comput (ProgramPart s),s,k) . (intloc 0 ) = s . (intloc 0 )

let k be Element of NAT ; :: thesis: (Comput (ProgramPart s),s,k) . (intloc 0 ) = s . (intloc 0 )
I +* (Start-At 0 ,SCM+FSA ) c= s +* (I +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:26;
then (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),k) . (intloc 0 ) = (s +* (I +* (Start-At 0 ,SCM+FSA ))) . (intloc 0 ) by Def4
.= s . (intloc 0 ) by A4, FUNCT_4:12 ;
hence (Comput (ProgramPart s),s,k) . (intloc 0 ) = s . (intloc 0 ) by A3, A17, Th41, SCMFSA10:92; :: thesis: verum

end;

end;