set S = s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ));
set SW = StepWhile=0 a,I,s;
defpred S₁[ Nat] means ((StepWhile=0 a,I,s) . $1) . a <> 0 ;
A3: (while=0 a,I) +* (Start-At 0 ,SCM+FSA ) c= s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )) by FUNCT_4:26;
consider f being Function of (product the Object-Kind of SCM+FSA ),NAT such that
A4: for k being Element of NAT holds
( f . ((StepWhile=0 a,I,s) . (k + 1)) < f . ((StepWhile=0 a,I,s) . k) or S₁[k] ) by A2, Def2;
deffunc H₁( Nat) -> Element of NAT = f . ((StepWhile=0 a,I,s) . $1);
A5: for k being Element of NAT holds
( H₁(k + 1) < H₁(k) or S₁[k] ) by A4;
consider m being Element of NAT such that
A6: S₁[m] and
A7: for n being Element of NAT st S₁[n] holds
m <= n from NAT_1:sch 18(A5);
take m ; :: thesis:
take m ; :: thesis:
thus m = m ; :: thesis:
thus ((StepWhile=0 a,I,s) . m) . a <> 0 by A6; :: thesis:
thus for n being Element of NAT st ((StepWhile=0 a,I,s) . n) . a <> 0 holds
m <= n by A7; :: thesis:
defpred S₂[ Nat] means ( $1 + 1 <= m implies ex k being Element of NAT st (StepWhile=0 a,I,s) . ($1 + 1) = Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),k );
A8: ProperBodyWhile=0 a,I,s by A1, Th19;

A9: now

let k be Element of NAT ; :: thesis:
assume A10: S₂[k] ; :: thesis:

now

set sk1 = (StepWhile=0 a,I,s) . (k + 1);
set sk = (StepWhile=0 a,I,s) . k;
assume A11: (k + 1) + 1 <= m ; :: thesis:
k + 0 < k + (1 + 1) by XREAL_1:8;
then k < m by A11, XXREAL_0:2;
then A12: ((StepWhile=0 a,I,s) . k) . a = 0 by A7;
(k + 1) + 0 < (k + 1) + 1 by XREAL_1:8;
then consider n being Element of NAT such that
A13: (StepWhile=0 a,I,s) . (k + 1) = Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),n by A10, A11, XXREAL_0:2;
A14: (StepWhile=0 a,I,s) . (k + 1) = Comput (ProgramPart (((StepWhile=0 a,I,s) . k) +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(((StepWhile=0 a,I,s) . k) +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((LifeSpan (ProgramPart (((StepWhile=0 a,I,s) . k) +* (I +* (Start-At 0 ,SCM+FSA )))),(((StepWhile=0 a,I,s) . k) +* (I +* (Start-At 0 ,SCM+FSA )))) + 3) by SCMFSA_9:def 4;
take m = n + ((LifeSpan (ProgramPart (((StepWhile=0 a,I,s) . (k + 1)) +* (I +* (Start-At 0 ,SCM+FSA )))),(((StepWhile=0 a,I,s) . (k + 1)) +* (I +* (Start-At 0 ,SCM+FSA )))) + 3); :: thesis:
( I is_closed_on (StepWhile=0 a,I,s) . k & I is_halting_on (StepWhile=0 a,I,s) . k ) by A8, A12, Def1;
then IC ((StepWhile=0 a,I,s) . (k + 1)) = 0 by A14, A12, SCMFSA_9:22;
hence (StepWhile=0 a,I,s) . ((k + 1) + 1) = Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),m by A13, SCMFSA_9:31; :: thesis:

end;

hence S₂[k + 1] ; :: thesis:

end;

A15: IC SCM+FSA in dom ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )) by SF_MASTR:65;
A16: S₂[ 0 ]

proof

assume 0 + 1 <= m ; :: thesis:
take n = (LifeSpan (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA )))) + 3; :: thesis:
thus (StepWhile=0 a,I,s) . (0 + 1) = Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),n by SCMFSA_9:30; :: thesis:

end;

A17: for k being Element of NAT holds S₂[k] from NAT_1:sch 1(A16, A9);

per cases ;

suppose A18: m = 0 ; :: thesis:

A19: DataPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))) = DataPart s by SCMFSA8A:11
.= DataPart ((StepWhile=0 a,I,s) . m) by A18, SCMFSA_9:def 4 ;
then (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))) . a = ((StepWhile=0 a,I,s) . m) . a by SCMFSA6A:38;
hence DataPart (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(LifeSpan (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))))) = DataPart ((StepWhile=0 a,I,s) . m) by A6, A3, A19, Th22; :: thesis:

end;

suppose A20: m <> 0 ; :: thesis:

set sm = (StepWhile=0 a,I,s) . m;
set sm1 = ((StepWhile=0 a,I,s) . m) +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ));
consider i being Nat such that
A21: m = i + 1 by A20, NAT_1:6;
reconsider i = i as Element of NAT by ORDINAL1:def 13;
set si = (StepWhile=0 a,I,s) . i;
A22: (StepWhile=0 a,I,s) . m = Comput (ProgramPart (((StepWhile=0 a,I,s) . i) +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(((StepWhile=0 a,I,s) . i) +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((LifeSpan (ProgramPart (((StepWhile=0 a,I,s) . i) +* (I +* (Start-At 0 ,SCM+FSA )))),(((StepWhile=0 a,I,s) . i) +* (I +* (Start-At 0 ,SCM+FSA )))) + 3) by A21, SCMFSA_9:def 4;
m = i + 1 by A21;
then consider n being Element of NAT such that
A23: (StepWhile=0 a,I,s) . m = Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),n by A17;
i < m by A21, NAT_1:13;
then A24: ((StepWhile=0 a,I,s) . i) . a = 0 by A7;
then ( I is_closed_on (StepWhile=0 a,I,s) . i & I is_halting_on (StepWhile=0 a,I,s) . i ) by A8, Def1;
then A25: IC ((StepWhile=0 a,I,s) . m) = 0 by A22, A24, SCMFSA_9:22;
A26: IC (((StepWhile=0 a,I,s) . m) +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))) = (((StepWhile=0 a,I,s) . m) +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))) . (IC SCM+FSA )
.= ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )) . (IC SCM+FSA ) by A15, FUNCT_4:14
.= IC ((StepWhile=0 a,I,s) . m) by A25, SF_MASTR:66 ;
ProgramPart ((StepWhile=0 a,I,s) . m) = ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))) by A23, AMI_1:123;
then ( DataPart (((StepWhile=0 a,I,s) . m) +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))) = DataPart ((StepWhile=0 a,I,s) . m) & ProgramPart (((StepWhile=0 a,I,s) . m) +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))) = ProgramPart ((StepWhile=0 a,I,s) . m) ) by FUNCT_4:100, SCMFSA8A:11;
then A27: ((StepWhile=0 a,I,s) . m) +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )) = (StepWhile=0 a,I,s) . m by A26, SCMFSA_9:29;
while=0 a,I is_halting_on (StepWhile=0 a,I,s) . m by A6, SCMFSA_9:18;
then ProgramPart (((StepWhile=0 a,I,s) . m) +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))) halts_on ((StepWhile=0 a,I,s) . m) +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )) by SCMFSA7B:def 8;
then consider j being Element of NAT such that
A28: CurInstr (ProgramPart ((StepWhile=0 a,I,s) . m)),(Comput (ProgramPart ((StepWhile=0 a,I,s) . m)),((StepWhile=0 a,I,s) . m),j) = halt SCM+FSA by A27, AMI_1:146;
x: Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(n + j) = Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),n),j by AMI_1:51;
T: ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))) = ProgramPart ((StepWhile=0 a,I,s) . m) by A23, AMI_1:123;
CurInstr (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(n + j)) = halt SCM+FSA by A23, A28, x, T;
then A29: Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(LifeSpan (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))) = Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(n + j) by AMI_1:127
.= Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),((StepWhile=0 a,I,s) . m),j by A23, AMI_1:51
.= Comput (ProgramPart ((StepWhile=0 a,I,s) . m)),((StepWhile=0 a,I,s) . m),(LifeSpan (ProgramPart ((StepWhile=0 a,I,s) . m)),((StepWhile=0 a,I,s) . m)) by A28, T, AMI_1:127 ;
(while=0 a,I) +* (Start-At 0 ,SCM+FSA ) c= (StepWhile=0 a,I,s) . m by A27, FUNCT_4:26;
hence DataPart (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(LifeSpan (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))))) = DataPart ((StepWhile=0 a,I,s) . m) by A6, A29, Th22; :: thesis:

end;

end;