set WHL = while<0 F4(),F5(),F3();
set iWHL = Initialized (stop (while<0 F4(),F5(),F3()));
set iI = Initialized (stop F3());
set s1 = F2() +* (Initialized (stop (while<0 F4(),F5(),F3())));
set ps = F2() | NAT ;
set sI = F2() +* (Initialized (stop F3()));
set m1 = (LifeSpan (F2() +* (Initialized (stop F3())))) + 2;
set s2 = (IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3())));
set m2 = LifeSpan ((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3()))));
A6: P1[ Dstate F2()] by A4;
A7: Initialized (stop F3()) c= F2() +* (Initialized (stop F3())) by FUNCT_4:26;
F3() is_closed_on F2() by A2, A4, A5;
then A8: F3() is_closed_on F2() +* (Initialized (stop F3())) by SCMPDS_6:38;
F3() is_halting_on F2() by A2, A4, A5;
then A9: ProgramPart (F2() +* (Initialized (stop F3()))) halts_on F2() +* (Initialized (stop F3())) by SCMPDS_6:def 3;
F2() +* (Initialized (stop F3())) = (F2() +* (Initialized (stop F3()))) +* (Initialized (stop F3())) by A7, FUNCT_4:79;
then ProgramPart ((F2() +* (Initialized (stop F3()))) +* (Initialized (stop F3()))) halts_on (F2() +* (Initialized (stop F3()))) +* (Initialized (stop F3())) by A9;
then A10: F3() is_halting_on F2() +* (Initialized (stop F3())) by SCMPDS_6:def 3;
set Es = IExec F3(),F2();
set bj = DataLoc ((IExec F3(),F2()) . F4()),F5();
deffunc H1( State of SCMPDS ) -> Element of NAT = F1($1);
A11: for t being State of SCMPDS st P1[ Dstate t] & t . F4() = F2() . F4() & t . (DataLoc (F2() . F4()),F5()) < 0 holds
( (IExec F3(),t) . F4() = t . F4() & F3() is_closed_on t & F3() is_halting_on t & H1( Dstate (IExec F3(),t)) < H1( Dstate t) & P1[ Dstate (IExec F3(),t)] ) by A5;
A12: for t being State of SCMPDS st P1[ Dstate t] & H1( Dstate t) = 0 holds
t . (DataLoc (F2() . F4()),F5()) >= 0 by A3;
( ( H1(F2()) = H1(F2()) or P1[F2()] ) & while<0 F4(),F5(),F3() is_closed_on F2() & while<0 F4(),F5(),F3() is_halting_on F2() ) from SCMPDS_8:sch 1(A1, A12, A6, A11);
 then A13: ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))) halts_on F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))) by SCMPDS_6:def 3;
deffunc H2( State of SCMPDS ) -> Element of NAT = F1($1);
A14: (IExec F3(),F2()) . F4() = F2() . F4() by A2, A4, A5;
then A15: for t being State of SCMPDS st P1[ Dstate t] & H2( Dstate t) = 0 holds
t . (DataLoc ((IExec F3(),F2()) . F4()),F5()) >= 0 by A3;
A16: for t being State of SCMPDS st P1[ Dstate t] & t . F4() = (IExec F3(),F2()) . F4() & t . (DataLoc ((IExec F3(),F2()) . F4()),F5()) < 0 holds
( (IExec F3(),t) . F4() = t . F4() & F3() is_closed_on t & F3() is_halting_on t & H2( Dstate (IExec F3(),t)) < H2( Dstate t) & P1[ Dstate (IExec F3(),t)] ) by A5, A14;
A17: P1[ Dstate (IExec F3(),F2())] by A2, A4, A5;
( ( H2( IExec F3(),F2()) = H2( IExec F3(),F2()) or P1[ IExec F3(),F2()] ) & while<0 F4(),F5(),F3() is_closed_on IExec F3(),F2() & while<0 F4(),F5(),F3() is_halting_on IExec F3(),F2() ) from SCMPDS_8:sch 1(A1, A15, A17, A16);
 then A18: ProgramPart ((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3())))) halts_on (IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3()))) by SCMPDS_6:def 3;
set s4 = Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),1;
set i1 = F4(),F5() >=0_goto ((card F3()) + 2);
set i2 = goto (- ((card F3()) + 1));
set b = DataLoc (F2() . F4()),F5();
A19: while<0 F4(),F5(),F3() = (F4(),F5() >=0_goto ((card F3()) + 2)) ';' (F3() ';' (goto (- ((card F3()) + 1)))) by SCMPDS_4:51;
set mI = LifeSpan (F2() +* (Initialized (stop F3())));
set s5 = Comput (ProgramPart (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),1)),(Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),1),(LifeSpan (F2() +* (Initialized (stop F3()))));
set l1 = (card F3()) + 1;
A20: IC (F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))) = 0 by SCMPDS_6:21;
A21: Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),(0 + 1) = Following (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),0 ) by AMI_1:14
.= Following (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))) by AMI_1:13
.= Exec (F4(),F5() >=0_goto ((card F3()) + 2)),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))) by A19, SCMPDS_6:22 ;
A22: DataPart (F2() +* (Initialized (stop F3()))) = DataPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))) by SCMPDS_4:24, SCMPDS_4:36;
now
let a be Int_position ; :: thesis: (F2() +* (Initialized (stop F3()))) . a = (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),1) . a
thus (F2() +* (Initialized (stop F3()))) . a = (F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))) . a by A22, SCMPDS_4:23
.= (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),1) . a by A21, SCMPDS_2:69 ; :: thesis: verum
end;
then A23: DataPart (F2() +* (Initialized (stop F3()))) = DataPart (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),1) by SCMPDS_4:23;
set m3 = (LifeSpan (F2() +* (Initialized (stop F3())))) + 1;
set s6 = Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),((LifeSpan (F2() +* (Initialized (stop F3())))) + 1);
(card F3()) + 1 < (card F3()) + 2 by XREAL_1:8;
then A24: (card F3()) + 1 in dom (while<0 F4(),F5(),F3()) by Th7;
thus ( F1(F2()) = F1(F2()) or P1[F2()] ) ; :: thesis: IExec (while<0 F4(),F5(),F3()),F2() = IExec (while<0 F4(),F5(),F3()),(IExec F3(),F2())
set m0 = LifeSpan (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))));
A25: dom (F2() | NAT ) = (dom F2()) /\ NAT by RELAT_1:90
.= (({(IC SCMPDS )} \/ SCM-Data-Loc ) \/ NAT ) /\ NAT by SCMPDS_4:19
.= NAT by XBOOLE_1:21 ;
A26: (IExec F3(),F2()) | NAT = ((Result (F2() +* (Initialized (stop F3())))) +* (F2() | NAT )) | NAT by SCMPDS_4:def 8
.= F2() | NAT by A25, FUNCT_4:24 ;
set s7 = Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),(((LifeSpan (F2() +* (Initialized (stop F3())))) + 1) + 1);
A27: IExec F3(),F2() = (Result (F2() +* (Initialized (stop F3())))) +* (F2() | NAT ) by SCMPDS_4:def 8;
A28: dom (F2() | NAT ) = NAT by SCMPDS_6:1;
( while<0 F4(),F5(),F3() c= Initialized (stop (while<0 F4(),F5(),F3())) & Initialized (stop (while<0 F4(),F5(),F3())) c= F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))) ) by FUNCT_4:26, SCMPDS_6:17;
then A29: while<0 F4(),F5(),F3() c= F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))) by XBOOLE_1:1;
Shift F3(),1 c= while<0 F4(),F5(),F3() by Lm2;
then Shift F3(),1 c= F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))) by A29, XBOOLE_1:1;
then A30: Shift F3(),1 c= Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),1 by AMI_1:81;
(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))) . (DataLoc ((F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))) . F4()),F5()) = (F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))) . (DataLoc (F2() . F4()),F5()) by SCMPDS_5:19
.= F2() . (DataLoc (F2() . F4()),F5()) by SCMPDS_5:19 ;
then A31: IC (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),1) = succ (IC (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))) by A2, A21, SCMPDS_2:69
.= 0 + 1 by A20 ;
then A32: IC (Comput (ProgramPart (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),1)),(Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),1),(LifeSpan (F2() +* (Initialized (stop F3()))))) = (card F3()) + 1 by A1, A7, A10, A8, A23, A30, SCMPDS_7:36;
Y: (ProgramPart (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),((LifeSpan (F2() +* (Initialized (stop F3())))) + 1))) /. (IC (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),((LifeSpan (F2() +* (Initialized (stop F3())))) + 1))) = (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),((LifeSpan (F2() +* (Initialized (stop F3())))) + 1)) . (IC (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),((LifeSpan (F2() +* (Initialized (stop F3())))) + 1))) by AMI_1:150;
ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))) = ProgramPart (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),1) by AMI_1:144;
then A33: Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),((LifeSpan (F2() +* (Initialized (stop F3())))) + 1) = Comput (ProgramPart (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),1)),(Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),1),(LifeSpan (F2() +* (Initialized (stop F3())))) by AMI_1:51;
then A34: CurInstr (ProgramPart (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),((LifeSpan (F2() +* (Initialized (stop F3())))) + 1))),(Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),((LifeSpan (F2() +* (Initialized (stop F3())))) + 1)) = (Comput (ProgramPart (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),1)),(Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),1),(LifeSpan (F2() +* (Initialized (stop F3()))))) . ((card F3()) + 1) by A1, A7, A10, A8, A31, A23, A30, SCMPDS_7:36, Y
.= (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),1) . ((card F3()) + 1) by AMI_1:54
.= (F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))) . ((card F3()) + 1) by AMI_1:54
.= (while<0 F4(),F5(),F3()) . ((card F3()) + 1) by A24, A29, GRFUNC_1:8
.= goto (- ((card F3()) + 1)) by Th8 ;
T: ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))) = ProgramPart (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),((LifeSpan (F2() +* (Initialized (stop F3())))) + 1)) by AMI_1:144;
A35: Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),(((LifeSpan (F2() +* (Initialized (stop F3())))) + 1) + 1) = Following (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),((LifeSpan (F2() +* (Initialized (stop F3())))) + 1)) by AMI_1:14
.= Exec (goto (- ((card F3()) + 1))),(Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),((LifeSpan (F2() +* (Initialized (stop F3())))) + 1)) by A34, T ;
then IC (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),(((LifeSpan (F2() +* (Initialized (stop F3())))) + 1) + 1)) = ICplusConst (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),((LifeSpan (F2() +* (Initialized (stop F3())))) + 1)),(0 - ((card F3()) + 1)) by SCMPDS_2:66
.= 0 by A32, A33, SCMPDS_7:1 ;
then A36: IC ((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3())))) = IC (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),((LifeSpan (F2() +* (Initialized (stop F3())))) + 2)) by SCMPDS_6:21;
A37: DataPart (Comput (ProgramPart (F2() +* (Initialized (stop F3())))),(F2() +* (Initialized (stop F3()))),(LifeSpan (F2() +* (Initialized (stop F3()))))) = DataPart (Comput (ProgramPart (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),1)),(Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),1),(LifeSpan (F2() +* (Initialized (stop F3()))))) by A1, A7, A10, A8, A31, A23, A30, SCMPDS_7:36;
now
let x be Int_position ; :: thesis: (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),(((LifeSpan (F2() +* (Initialized (stop F3())))) + 1) + 1)) . x = ((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3())))) . x
A38: not x in dom (Initialized (stop (while<0 F4(),F5(),F3()))) by SCMPDS_4:31;
A39: not x in dom (F2() | NAT ) by A28, SCMPDS_2:53;
(Comput (ProgramPart (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),1)),(Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),1),(LifeSpan (F2() +* (Initialized (stop F3()))))) . x = (Comput (ProgramPart (F2() +* (Initialized (stop F3())))),(F2() +* (Initialized (stop F3()))),(LifeSpan (F2() +* (Initialized (stop F3()))))) . x by A37, SCMPDS_4:23
.= (Result (F2() +* (Initialized (stop F3())))) . x by A9, AMI_1:122
.= (IExec F3(),F2()) . x by A27, A39, FUNCT_4:12 ;
hence (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),(((LifeSpan (F2() +* (Initialized (stop F3())))) + 1) + 1)) . x = (IExec F3(),F2()) . x by A33, A35, SCMPDS_2:66
.= ((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3())))) . x by A38, FUNCT_4:12 ;
:: thesis: verum
end;
then A40: DataPart (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),(((LifeSpan (F2() +* (Initialized (stop F3())))) + 1) + 1)) = DataPart ((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3())))) by SCMPDS_4:23;
((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3())))) | NAT = (((Result (F2() +* (Initialized (stop F3())))) +* (F2() | NAT )) +* (Initialized (stop (while<0 F4(),F5(),F3())))) | NAT by SCMPDS_4:def 8
.= (((Result (F2() +* (Initialized (stop F3())))) +* (F2() | NAT )) | NAT ) +* ((Initialized (stop (while<0 F4(),F5(),F3()))) | NAT ) by FUNCT_4:75
.= (F2() | NAT ) +* ((Initialized (stop (while<0 F4(),F5(),F3()))) | NAT ) by A25, FUNCT_4:24
.= (F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))) | NAT by FUNCT_4:75
.= (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),((LifeSpan (F2() +* (Initialized (stop F3())))) + 2)) | NAT by SCMPDS_7:6 ;
then A41: Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),((LifeSpan (F2() +* (Initialized (stop F3())))) + 2) = (IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3()))) by A40, A36, SCMPDS_7:7;
then CurInstr (ProgramPart (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),((LifeSpan (F2() +* (Initialized (stop F3())))) + 2))),(Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),((LifeSpan (F2() +* (Initialized (stop F3())))) + 2)) = F4(),F5() >=0_goto ((card F3()) + 2) by A19, SCMPDS_6:22;
then LifeSpan (F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))) > (LifeSpan (F2() +* (Initialized (stop F3())))) + 2 by A13, SCMPDS_6:2, SCMPDS_6:31;
then consider nn being Nat such that
A42: LifeSpan (F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))) = ((LifeSpan (F2() +* (Initialized (stop F3())))) + 2) + nn by NAT_1:10;
reconsider nn = nn as Element of NAT by ORDINAL1:def 13;
T: ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))) = ProgramPart ((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3())))) by A41, AMI_1:144;
Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),(((LifeSpan (F2() +* (Initialized (stop F3())))) + 2) + (LifeSpan ((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3())))))) = Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3())))),(LifeSpan ((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3()))))) by A41, AMI_1:51;
then CurInstr (ProgramPart (Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),(((LifeSpan (F2() +* (Initialized (stop F3())))) + 2) + (LifeSpan ((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3())))))))),(Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),(((LifeSpan (F2() +* (Initialized (stop F3())))) + 2) + (LifeSpan ((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3()))))))) = halt SCMPDS by A18, AMI_1:def 46, T;
then ((LifeSpan (F2() +* (Initialized (stop F3())))) + 2) + (LifeSpan ((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3()))))) >= LifeSpan (F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))) by A13, AMI_1:def 46;
then A43: LifeSpan ((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3())))) >= nn by A42, XREAL_1:8;
A44: Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),(F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))),(LifeSpan (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))) = Comput (ProgramPart (F2() +* (Initialized (stop (while<0 F4(),F5(),F3()))))),((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3())))),nn by A41, A42, AMI_1:51;
then CurInstr (ProgramPart (Comput (ProgramPart ((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3()))))),((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3())))),nn)),(Comput (ProgramPart ((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3()))))),((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3())))),nn) = halt SCMPDS by A13, AMI_1:def 46, T;
then nn >= LifeSpan ((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3())))) by A18, AMI_1:def 46;
then nn = LifeSpan ((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3())))) by A43, XXREAL_0:1;
then Result (F2() +* (Initialized (stop (while<0 F4(),F5(),F3())))) = Comput (ProgramPart ((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3()))))),((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3())))),(LifeSpan ((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3()))))) by A13, A44, AMI_1:122, T;
hence IExec (while<0 F4(),F5(),F3()),F2() = (Comput (ProgramPart ((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3()))))),((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3())))),(LifeSpan ((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3())))))) +* (F2() | NAT ) by SCMPDS_4:def 8
.= (Result ((IExec F3(),F2()) +* (Initialized (stop (while<0 F4(),F5(),F3()))))) +* ((IExec F3(),F2()) | NAT ) by A18, A26, AMI_1:122
.= IExec (while<0 F4(),F5(),F3()),(IExec F3(),F2()) by SCMPDS_4:def 8 ;
:: thesis: verum