set S = s +* ((while>0 a,I) +* (Start-At (insloc 0 )));
set SW = StepWhile>0 a,I,s;
defpred S₁[ Element of NAT ] means ((StepWhile>0 a,I,s) . $1) . a <= 0 ;
A3: (while>0 a,I) +* (Start-At (insloc 0 )) c= s +* ((while>0 a,I) +* (Start-At (insloc 0 ))) 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, Def5;
deffunc H₁( Element of 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 SEQ_4:sch 1(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[ Element of NAT ] means ( $1 + 1 <= m implies ex k being Element of NAT st (StepWhile>0 a,I,s) . ($1 + 1) = Computation (s +* ((while>0 a,I) +* (Start-At (insloc 0 )))),k );
A8: ProperBodyWhile>0 a,I,s by A1, Th32;

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) = Computation (s +* ((while>0 a,I) +* (Start-At (insloc 0 )))),n by A10, A11, XXREAL_0:2;
A14: (StepWhile>0 a,I,s) . (k + 1) = Computation (((StepWhile>0 a,I,s) . k) +* ((while>0 a,I) +* (Start-At (insloc 0 )))),((LifeSpan (((StepWhile>0 a,I,s) . k) +* (I +* (Start-At (insloc 0 ))))) + 3) by SCMFSA_9:def 5;
take m = n + ((LifeSpan (((StepWhile>0 a,I,s) . (k + 1)) +* (I +* (Start-At (insloc 0 ))))) + 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, Def4;
then IC ((StepWhile>0 a,I,s) . (k + 1)) = insloc 0 by A14, A12, SCMFSA_9:47;
hence (StepWhile>0 a,I,s) . ((k + 1) + 1) = Computation (s +* ((while>0 a,I) +* (Start-At (insloc 0 )))),m by A13, SCMFSA_9:52; :: thesis:

end;

hence S₂[k + 1] ; :: thesis:

end;

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

proof

assume 0 + 1 <= m ; :: thesis:
take n = (LifeSpan (s +* (I +* (Start-At (insloc 0 ))))) + 3; :: thesis:
thus (StepWhile>0 a,I,s) . (0 + 1) = Computation (s +* ((while>0 a,I) +* (Start-At (insloc 0 )))),n by SCMFSA_9:51; :: 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 (insloc 0 )))) = DataPart s by SCMFSA8A:11
.= DataPart ((StepWhile>0 a,I,s) . m) by A18, SCMFSA_9:def 5 ;
then (s +* ((while>0 a,I) +* (Start-At (insloc 0 )))) . a = ((StepWhile>0 a,I,s) . m) . a by SCMFSA6A:38;
hence DataPart (Computation (s +* ((while>0 a,I) +* (Start-At (insloc 0 )))),(LifeSpan (s +* ((while>0 a,I) +* (Start-At (insloc 0 )))))) = DataPart ((StepWhile>0 a,I,s) . m) by A6, A3, A19, Th35; :: 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 (insloc 0 )));
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 = Computation (((StepWhile>0 a,I,s) . i) +* ((while>0 a,I) +* (Start-At (insloc 0 )))),((LifeSpan (((StepWhile>0 a,I,s) . i) +* (I +* (Start-At (insloc 0 ))))) + 3) by A21, SCMFSA_9:def 5;
m = i + 1 by A21;
then consider n being Element of NAT such that
A23: (StepWhile>0 a,I,s) . m = Computation (s +* ((while>0 a,I) +* (Start-At (insloc 0 )))),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, Def4;
then A25: IC ((StepWhile>0 a,I,s) . m) = insloc 0 by A22, A24, SCMFSA_9:47;
A26: IC (((StepWhile>0 a,I,s) . m) +* ((while>0 a,I) +* (Start-At (insloc 0 )))) = (((StepWhile>0 a,I,s) . m) +* ((while>0 a,I) +* (Start-At (insloc 0 )))) . (IC SCM+FSA ) by AMI_1:def 15
.= ((while>0 a,I) +* (Start-At (insloc 0 ))) . (IC SCM+FSA ) by A15, FUNCT_4:14
.= IC ((StepWhile>0 a,I,s) . m) by A25, SF_MASTR:66 ;
((StepWhile>0 a,I,s) . m) | NAT = (s +* ((while>0 a,I) +* (Start-At (insloc 0 )))) | NAT by A23, AMI_1:123;
then ( DataPart (((StepWhile>0 a,I,s) . m) +* ((while>0 a,I) +* (Start-At (insloc 0 )))) = DataPart ((StepWhile>0 a,I,s) . m) & (((StepWhile>0 a,I,s) . m) +* ((while>0 a,I) +* (Start-At (insloc 0 )))) | NAT = ((StepWhile>0 a,I,s) . m) | NAT ) by FUNCT_4:100, SCMFSA8A:11;
then A27: ((StepWhile>0 a,I,s) . m) +* ((while>0 a,I) +* (Start-At (insloc 0 ))) = (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_9:43;
then ((StepWhile>0 a,I,s) . m) +* ((while>0 a,I) +* (Start-At (insloc 0 ))) is halting by SCMFSA7B:def 8;
then consider j being Element of NAT such that
A28: CurInstr (Computation ((StepWhile>0 a,I,s) . m),j) = halt SCM+FSA by A27, AMI_1:def 20;
CurInstr (Computation (s +* ((while>0 a,I) +* (Start-At (insloc 0 )))),(n + j)) = halt SCM+FSA by A23, A28, AMI_1:51;
then A29: Computation (s +* ((while>0 a,I) +* (Start-At (insloc 0 )))),(LifeSpan (s +* ((while>0 a,I) +* (Start-At (insloc 0 ))))) = Computation (s +* ((while>0 a,I) +* (Start-At (insloc 0 )))),(n + j) by AMI_1:127
.= Computation ((StepWhile>0 a,I,s) . m),j by A23, AMI_1:51
.= Computation ((StepWhile>0 a,I,s) . m),(LifeSpan ((StepWhile>0 a,I,s) . m)) by A28, AMI_1:127 ;
(while>0 a,I) +* (Start-At (insloc 0 )) c= (StepWhile>0 a,I,s) . m by A27, FUNCT_4:26;
hence DataPart (Computation (s +* ((while>0 a,I) +* (Start-At (insloc 0 )))),(LifeSpan (s +* ((while>0 a,I) +* (Start-At (insloc 0 )))))) = DataPart ((StepWhile>0 a,I,s) . m) by A6, A29, Th35; :: thesis:

end;