let s be State of SCMPDS ; :: thesis:
set SA0 = Start-At (inspos 0 );
let i be parahalting Instruction of SCMPDS ; :: thesis:
set Li = Load i;
set Mi = stop (Load i);
set sI = s +* (Initialized (stop (Load i)));
set Is = Initialized s;
set IC1 = IC (Computation (s +* (Initialized (stop (Load i)))),1);
A1: s +* (Initialized (stop (Load i))) is halting by FUNCT_4:26, SCMPDS_4:63;
A2: inspos 1 in dom (Initialized (stop (Load i))) by Th13;
A3: inspos 0 in dom (Initialized (stop (Load i))) by Th13;

A4: now

set Y = NAT ;
set X = SCM-Data-Loc \/ {(IC SCMPDS )};
assume A5: Result (s +* (Initialized (stop (Load i)))) = Exec i,(s +* (Initialized (stop (Load i)))) ; :: thesis:
s +* (Initialized (stop (Load i))) = (s +* (stop (Load i))) +* (Start-At (inspos 0 )) by FUNCT_4:15;
then Initialized s,s +* (Initialized (stop (Load i))) equal_outside NAT by AMI_1:120, FUNCT_7:106;
then (Initialized s) | (SCM-Data-Loc \/ {(IC SCMPDS )}) = (s +* (Initialized (stop (Load i)))) | (SCM-Data-Loc \/ {(IC SCMPDS )}) by Th41;
then A6: (Exec i,(Initialized s)) | (SCM-Data-Loc \/ {(IC SCMPDS )}) = (Exec i,(s +* (Initialized (stop (Load i))))) | (SCM-Data-Loc \/ {(IC SCMPDS )}) by SCMPDS_3:7;
A7: NAT /\ ((SCM-Data-Loc \/ {(IC SCMPDS )}) \/ NAT ) c= NAT /\ ((SCM-Data-Loc \/ {(IC SCMPDS )}) \/ NAT ) ;
A8: dom (IExec (Load i),s) = the carrier of SCMPDS by AMI_1:79;
A9: dom (Exec i,(Initialized s)) = the carrier of SCMPDS by AMI_1:79;
A10: dom s = (SCM-Data-Loc \/ {(IC SCMPDS )}) \/ NAT by AMI_1:79, SCMPDS_3:5;

now

thus dom ((Exec i,(Initialized s)) | NAT ) = (dom s) /\ NAT by A9, A10, RELAT_1:90, SCMPDS_3:5; :: thesis:
let x be set ; :: thesis:
assume x in dom ((Exec i,(Initialized s)) | NAT ) ; :: thesis:
then A11: x in NAT /\ ((SCM-Data-Loc \/ {(IC SCMPDS )}) \/ NAT ) by A9, RELAT_1:90, SCMPDS_3:5;
then x is Element of NAT by XBOOLE_1:21;
then reconsider x' = x as Instruction-Location of SCMPDS by AMI_1:def 4;
x in NAT by A11, XBOOLE_1:21;
hence ((Exec i,(Initialized s)) | NAT ) . x = (Exec i,(Initialized s)) . x by FUNCT_1:72
.= (Initialized s) . x' by AMI_1:def 13
.= s . x by Th40 ;
:: thesis:

end;

then A12: (Exec i,(Initialized s)) | NAT = s | NAT by FUNCT_1:68;
dom (Exec i,(s +* (Initialized (stop (Load i))))) = the carrier of SCMPDS by AMI_1:79;
then A13: (IExec (Load i),s) | NAT = s | NAT by A5, A10, A7, FUNCT_4:93, SCMPDS_3:5;
SCM-Data-Loc \/ {(IC SCMPDS )} misses NAT

proof

assume SCM-Data-Loc \/ {(IC SCMPDS )} meets NAT ; :: thesis:
then consider x being set such that
A14: x in SCM-Data-Loc \/ {(IC SCMPDS )} and
A15: x in NAT by XBOOLE_0:3;
A16: ( x in SCM-Data-Loc or x in {(IC SCMPDS )} ) by A14, XBOOLE_0:def 3;

per cases by A16, TARSKI:def 1;

suppose x in SCM-Data-Loc ; :: thesis:

hence contradiction by A15, AMI_2:29, XBOOLE_0:3; :: thesis:

end;

suppose x = IC SCMPDS ; :: thesis:

then reconsider l = IC SCMPDS as Instruction-Location of SCMPDS by A15, AMI_1:def 4;
l = IC SCMPDS ;
hence contradiction by AMI_1:48; :: thesis:

end;

then SCM-Data-Loc \/ {(IC SCMPDS )} misses dom (s | NAT ) by RELAT_1:87, XBOOLE_1:63;
then (IExec (Load i),s) | (SCM-Data-Loc \/ {(IC SCMPDS )}) = (Exec i,(s +* (Initialized (stop (Load i))))) | (SCM-Data-Loc \/ {(IC SCMPDS )}) by A5, FUNCT_4:76;
then A17: (Exec i,(Initialized s)) | ((SCM-Data-Loc \/ {(IC SCMPDS )}) \/ NAT ) = (IExec (Load i),s) | ((SCM-Data-Loc \/ {(IC SCMPDS )}) \/ NAT ) by A6, A13, A12, RELAT_1:185;
thus Exec i,(Initialized s) = (Exec i,(Initialized s)) | ((SCM-Data-Loc \/ {(IC SCMPDS )}) \/ NAT ) by A9, RELAT_1:98, SCMPDS_3:5
.= IExec (Load i),s by A8, A17, RELAT_1:98, SCMPDS_3:5 ; :: thesis:

end;

A18: (Initialized (stop (Load i))) . (inspos 1) = halt SCMPDS by Th13;
A19: (Initialized (stop (Load i))) . (inspos 0 ) = i by Th13;
A20: Initialized (stop (Load i)) c= s +* (Initialized (stop (Load i))) by FUNCT_4:26;
then A21: IC (Computation (s +* (Initialized (stop (Load i)))),1) in dom (stop (Load i)) by SCMPDS_4:def 9;
A22: Computation (s +* (Initialized (stop (Load i)))),(0 + 1) = Following (Computation (s +* (Initialized (stop (Load i)))),0 ) by AMI_1:14
.= Following (s +* (Initialized (stop (Load i)))) by AMI_1:13
.= Exec ((s +* (Initialized (stop (Load i)))) . (inspos 0 )),(s +* (Initialized (stop (Load i)))) by Th18, FUNCT_4:26
.= Exec i,(s +* (Initialized (stop (Load i)))) by A3, A19, A20, GRFUNC_1:8 ;

per cases by A21, Th11;

suppose A23: IC (Computation (s +* (Initialized (stop (Load i)))),1) = inspos 0 ; :: thesis:

set Ni = InsCode i;
Next (IC (s +* (Initialized (stop (Load i))))) = Next (inspos 0 ) by Th18, FUNCT_4:26
.= inspos 1 ;
then A24: InsCode i in {0 ,1,4,5,6} by A22, A23, SCMPDS_4:6;
A25: CurInstr (Computation (s +* (Initialized (stop (Load i)))),1) = (s +* (Initialized (stop (Load i)))) . (inspos 0 ) by A22, A23, AMI_1:def 13
.= i by A3, A19, A20, GRFUNC_1:8 ;
A26: InsCode i <> 1 by Th26;

hereby :: thesis:

per cases ;

suppose i = halt SCMPDS ; :: thesis:

hence Exec i,(Initialized s) = IExec (Load i),s by A1, A4, A22, A25, AMI_1:def 22; :: thesis:

end;

suppose A27: i <> halt SCMPDS ; :: thesis:

A28: for loc being Instruction-Location of SCMPDS holds (s +* (Initialized (stop (Load i)))) . loc = (Exec i,(s +* (Initialized (stop (Load i))))) . loc by AMI_1:def 13;

A29: now

let b be Int_position ; :: thesis:

per cases by A24, A26, ENUMSET1:def 3;

suppose InsCode i = 0 ; :: thesis:

then ex k1 being Integer st i = goto k1 by SCMPDS_2:35;
hence (s +* (Initialized (stop (Load i)))) . b = (Exec i,(s +* (Initialized (stop (Load i))))) . b by SCMPDS_2:66; :: thesis:

end;

suppose InsCode i = 4 ; :: thesis:

then ex a being Int_position ex k1, k2 being Integer st i = a,k1 <>0_goto k2 by SCMPDS_2:39;
hence (s +* (Initialized (stop (Load i)))) . b = (Exec i,(s +* (Initialized (stop (Load i))))) . b by SCMPDS_2:67; :: thesis:

end;

suppose InsCode i = 5 ; :: thesis:

then ex a being Int_position ex k1, k2 being Integer st i = a,k1 <=0_goto k2 by SCMPDS_2:40;
hence (s +* (Initialized (stop (Load i)))) . b = (Exec i,(s +* (Initialized (stop (Load i))))) . b by SCMPDS_2:68; :: thesis:

end;

suppose InsCode i = 6 ; :: thesis:

then ex a being Int_position ex k1, k2 being Integer st i = a,k1 >=0_goto k2 by SCMPDS_2:41;
hence (s +* (Initialized (stop (Load i)))) . b = (Exec i,(s +* (Initialized (stop (Load i))))) . b by SCMPDS_2:69; :: thesis:

end;

A30: Following (s +* (Initialized (stop (Load i)))) = Following (Computation (s +* (Initialized (stop (Load i)))),0 ) by AMI_1:13
.= Exec i,(s +* (Initialized (stop (Load i)))) by A22, AMI_1:14 ;
A31: IC (s +* (Initialized (stop (Load i)))) = IC (Exec i,(s +* (Initialized (stop (Load i))))) by A22, A23, Th18, FUNCT_4:26;
then A32: s +* (Initialized (stop (Load i))) = Exec i,(s +* (Initialized (stop (Load i)))) by A29, A28, SCMPDS_2:54;

now

let n be Element of NAT ; :: thesis:
Computation (s +* (Initialized (stop (Load i)))),n = s +* (Initialized (stop (Load i))) by A31, A29, A28, A30, AMI_1:130, SCMPDS_2:54
.= Following (Computation (s +* (Initialized (stop (Load i)))),0 ) by A32, A30, AMI_1:13
.= Computation (s +* (Initialized (stop (Load i)))),(0 + 1) by AMI_1:14 ;
hence CurInstr (Computation (s +* (Initialized (stop (Load i)))),n) <> halt SCMPDS by A25, A27; :: thesis:

end;

then not s +* (Initialized (stop (Load i))) is halting by AMI_1:def 20;
hence Exec i,(Initialized s) = IExec (Load i),s by FUNCT_4:26, SCMPDS_4:63; :: thesis:

end;

suppose IC (Computation (s +* (Initialized (stop (Load i)))),1) = inspos 1 ; :: thesis:

then CurInstr (Computation (s +* (Initialized (stop (Load i)))),1) = (s +* (Initialized (stop (Load i)))) . (inspos 1) by A22, AMI_1:def 13
.= halt SCMPDS by A2, A18, A20, GRFUNC_1:8 ;
hence Exec i,(Initialized s) = IExec (Load i),s by A1, A4, A22, AMI_1:def 22; :: thesis:

end;