reconsider t14 = ((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0))) as InitHalting good Program of SCM+FSA ;
reconsider t16 = (t14 ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1)))) as good Program of SCM+FSA ;
reconsider Wt = while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))) as InitHalting good Program of SCM+FSA by Lm4, Th11;
let s be State of SCM+FSA; :: thesis:
let P be Instruction-Sequence of SCM+FSA; :: thesis:
assume A1: s . (intloc (0 + 1)) = (len (s . (fsloc 0))) - 1 ; :: thesis:
A2: t16 ";" Wt is good Program of SCM+FSA ;

per cases ;

suppose A3: len (s . (fsloc 0)) <= 1 ; :: thesis:

hence s . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))))),P,s)) . (fsloc 0) are_fiberwise_equipotent by A1, Lm11, SCM_HALT:67, XREAL_1:47; :: thesis:

now :: thesis:

let i, j be Nat; :: thesis:
assume that
A4: i >= 1 and
A5: j <= len (s . (fsloc 0)) and
A6: i < j ; :: thesis:
j <= 1 by A3, A5, XXREAL_0:2;
hence contradiction by A4, A6, XXREAL_0:2; :: thesis:

end;

hence for i, j being Nat st i >= 1 & j <= len (s . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))))),P,s)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))))),P,s)) . (fsloc 0)) . j holds
x1 >= x2 ; :: thesis:

end;

suppose len (s . (fsloc 0)) > 1 ; :: thesis:

then 1 - 1 < (len (s . (fsloc 0))) - 1 by XREAL_1:9;
then reconsider m = (len (s . (fsloc 0))) - 1 as Element of NAT by INT_1:3;
defpred S₁[ Nat] means for t being State of SCM+FSA
for Q being Instruction-Sequence of SCM+FSA st t . (intloc (0 + 1)) = $1 & t . (intloc (0 + 1)) <= (len (t . (fsloc 0))) - 1 & ( for i, j being Nat st i >= 1 & j <= (len (t . (fsloc 0))) - (t . (intloc (0 + 1))) & i < j holds
for x1, x2 being Integer st x1 = (t . (fsloc 0)) . i & x2 = (t . (fsloc 0)) . j holds
x1 >= x2 ) holds
( t . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))))),Q,t)) . (fsloc 0) are_fiberwise_equipotent & ( for i, j being Nat st i >= 1 & j <= len (t . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))))),Q,t)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))))),Q,t)) . (fsloc 0)) . j holds
x1 >= x2 ) );

A7: now :: thesis:

let k be Nat; :: thesis:
assume A8: S₁[k] ; :: thesis:

now :: thesis:

let t be State of SCM+FSA; :: thesis:
let Q be Instruction-Sequence of SCM+FSA; :: thesis:
assume that
A9: t . (intloc (0 + 1)) = k + 1 and
A10: t . (intloc (0 + 1)) <= (len (t . (fsloc 0))) - 1 ; :: thesis:
len (t . (fsloc 0)) < (len (t . (fsloc 0))) + (t . (intloc (0 + 1))) by A9, XREAL_1:29;
then A11: (len (t . (fsloc 0))) - (t . (intloc (0 + 1))) < ((len (t . (fsloc 0))) + (t . (intloc (0 + 1)))) - (t . (intloc (0 + 1))) by XREAL_1:9;
then A12: ((len (t . (fsloc 0))) - (t . (intloc (0 + 1)))) + 1 <= len (t . (fsloc 0)) by INT_1:7;
- ((len (t . (fsloc 0))) - 1) <= - (t . (intloc (0 + 1))) by A10, XREAL_1:24;
then A13: (len (t . (fsloc 0))) + (- ((len (t . (fsloc 0))) - 1)) <= (len (t . (fsloc 0))) + (- (t . (intloc (0 + 1)))) by XREAL_1:6;
then reconsider k1 = (len (t . (fsloc 0))) - (t . (intloc (0 + 1))) as Element of NAT by INT_1:3;
set IW = IExec ((t16 ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))),Q,t);
set ts = IExec (t16,Q,t);
set B1 = SubFrom ((intloc (0 + 1)),(intloc 0));
set IB = IExec (((((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),Q,t);
A14: (IExec (t16,Q,t)) . (fsloc 0) = t . (fsloc 0) by Lm20;
A15: (IExec (t16,Q,t)) . (intloc (1 + 1)) = (len (t . (fsloc 0))) - (t . (intloc (0 + 1))) by Lm20;
then (IExec (t16,Q,t)) . (intloc (1 + 1)) <= len ((IExec (t16,Q,t)) . (fsloc 0)) by A11, Lm20;
then consider n being Nat, x1 being Integer such that
A16: n = ((IExec ((while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))))),Q,(IExec (t16,Q,t)))) . (intloc (3 + 1))) - ((IExec (t16,Q,t)) . (intloc (3 + 1))) and
A17: n <= k1 and
A18: ( k1 - n >= 1 implies ( x1 = ((IExec (t16,Q,t)) . (fsloc 0)) . (k1 - n) & x1 >= (IExec (t16,Q,t)) . (intloc (5 + 1)) ) ) and
A19: for i being Nat st i > k1 - n & i < k1 + 1 holds
ex x2 being Integer st
( x2 = ((IExec (t16,Q,t)) . (fsloc 0)) . i & x2 <= (IExec (t16,Q,t)) . (intloc (5 + 1)) ) by A15, A13, Lm17;
A20: k1 < k1 + 1 by XREAL_1:29;
then A21: n < k1 + 1 by A17, XXREAL_0:2;
then A22: n - n < (k1 + 1) - n by XREAL_1:9;
reconsider n3 = (t . (fsloc 0)) . (k1 + 1) as Integer ;
A23: 1 + 0 <= k1 + 1 by INT_1:7;
A24: (IExec (((((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),Q,t)) . (fsloc 0) = (Exec ((SubFrom ((intloc (0 + 1)),(intloc 0))),(IExec ((((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))),Q,t)))) . (fsloc 0) by Lm11, SCM_HALT:24
.= (IExec ((((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))),Q,t)) . (fsloc 0) by SCMFSA_2:65
.= (IExec ((Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1)))))),Q,(IExec ((t16 ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))),Q,t)))) . (fsloc 0) by A2, Lm10, SCM_HALT:21 ;
set mm = (k1 + 1) - n;
|.(k1 + 1).| = k1 + 1 by ABSVALUE:def 1;
then A25: (IExec (t16,Q,t)) . (intloc (5 + 1)) = (t . (fsloc 0)) /. (k1 + 1) by Lm20
.= n3 by A12, A23, FINSEQ_4:15 ;
then A26: ( k1 - n >= 1 implies ( x1 = (t . (fsloc 0)) . (k1 - n) & x1 >= n3 ) ) by A18, Lm20;
A27: (IExec (t16,Q,t)) . (intloc (1 + 1)) = k1 by Lm20;
A28: (IExec ((t16 ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))),Q,t)) . (fsloc 0) = (IExec ((while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))))),Q,(IExec (t16,Q,t)))) . (fsloc 0) by Lm5, SCM_HALT:21
.= t . (fsloc 0) by A14, A11, A27, Lm17 ;
A29: (IExec ((t16 ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))),Q,t)) . (intloc (2 + 1)) = (IExec ((while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0))))))))),Q,(IExec (t16,Q,t)))) . (intloc (2 + 1)) by Lm5, SCM_HALT:20
.= (IExec (t16,Q,t)) . (intloc (2 + 1)) by A14, A11, A27, Lm17
.= k1 + 1 by Lm20 ;
then A30: (IExec ((t16 ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))),Q,t)) . (intloc (2 + 1)) <= len ((IExec ((t16 ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))),Q,t)) . (fsloc 0)) by A11, A28, INT_1:7;
A31: (IExec ((t16 ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))),Q,t)) . (intloc (3 + 1)) = n + ((IExec (t16,Q,t)) . (intloc (3 + 1))) by A16, Lm5, SCM_HALT:20
.= n + 0 by Lm20
.= n ;
then A32: (IExec ((t16 ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))),Q,t)) . (fsloc 0),(IExec ((Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1)))))),Q,(IExec ((t16 ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))),Q,t)))) . (fsloc 0) are_fiberwise_equipotent by A28, A29, A21, A12, Lm19;
A33: n < (IExec ((t16 ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))),Q,t)) . (intloc (2 + 1)) by A17, A29, A20, XXREAL_0:2;
then (IExec ((t16 ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))),Q,t)) . (fsloc 0),(IExec (((((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),Q,t)) . (fsloc 0) are_fiberwise_equipotent by A28, A24, A31, A29, A12, Lm19;
then A34: len ((IExec (((((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),Q,t)) . (fsloc 0)) = len (t . (fsloc 0)) by A28, RFINSEQ:3;
A35: (IExec (((((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),Q,t)) . (intloc (0 + 1)) = (k + 1) - 1 by A9, Lm9, Lm11, SCM_HALT:66
.= k ;
then (IExec (((((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),Q,t)) . (intloc (0 + 1)) < t . (intloc (0 + 1)) by A9, XREAL_1:29;
then A36: (IExec (((((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),Q,t)) . (intloc (0 + 1)) <= (len ((IExec (((((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),Q,t)) . (fsloc 0))) - 1 by A10, A34, XXREAL_0:2;
A37: ((IExec ((Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1)))))),Q,(IExec ((t16 ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))),Q,t)))) . (fsloc 0)) . ((k1 + 1) - n) = ((IExec ((t16 ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))),Q,t)) . (fsloc 0)) . (k1 + 1) by A28, A31, A29, A21, A12, Lm19;

hereby :: thesis:

A38: (IExec ((Times ((intloc (0 + 1)),(((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))))),Q,t)) . (fsloc 0) = (IExec ((Times ((intloc (0 + 1)),(((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))))),Q,(IExec (((((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),Q,t)))) . (fsloc 0) by A9, Lm9, Lm11, SCM_HALT:69;
assume A39: for i, j being Nat st i >= 1 & j <= (len (t . (fsloc 0))) - (t . (intloc (0 + 1))) & i < j holds
for x1, x2 being Integer st x1 = (t . (fsloc 0)) . i & x2 = (t . (fsloc 0)) . j holds
x1 >= x2 ; :: thesis:

A40: now :: thesis:

A41: (k1 + 1) - n <= (k1 + 1) - 0 by XREAL_1:13;
then (k1 - n) + 1 <= k1 + 1 ;
then A42: k1 - n <= k1 by XREAL_1:6;
let i, j be Nat; :: thesis:
assume that
A43: i >= 1 and
A44: j <= (len ((IExec (((((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),Q,t)) . (fsloc 0))) - ((IExec (((((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),Q,t)) . (intloc (0 + 1))) and
A45: i < j ; :: thesis:
A46: 1 <= j by A43, A45, XXREAL_0:2;
(len ((IExec (((((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),Q,t)) . (fsloc 0))) - ((IExec (((((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),Q,t)) . (intloc (0 + 1))) = k1 + 1 by A9, A35, A34;
then A47: j - 1 <= k1 by A44, XREAL_1:20;
A48: 1 - 1 <= i - 1 by A43, XREAL_1:9;
then reconsider i1 = i - 1 as Element of NAT by INT_1:3;
A49: i - 1 < j - 1 by A45, XREAL_1:9;
then reconsider j1 = j - 1 as Element of NAT by A48, INT_1:3;
let y1, y2 be Integer; :: thesis:
assume that
A50: y1 = ((IExec (((((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),Q,t)) . (fsloc 0)) . i and
A51: y2 = ((IExec (((((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),Q,t)) . (fsloc 0)) . j ; :: thesis:

per cases by XXREAL_0:1;

suppose A52: i < (k1 + 1) - n ; :: thesis:

then A53: y1 = (t . (fsloc 0)) . i by A28, A24, A31, A29, A21, A30, A43, A50, Lm19;

hereby :: thesis:

per cases by XXREAL_0:1;

suppose A54: j < (k1 + 1) - n ; :: thesis:

then j < k1 + 1 by A41, XXREAL_0:2;
then A55: j <= k1 by INT_1:7;
y2 = (t . (fsloc 0)) . j by A28, A24, A31, A29, A21, A30, A51, A46, A54, Lm19;
hence y1 >= y2 by A39, A43, A45, A53, A55; :: thesis:

end;

suppose A56: j > (k1 + 1) - n ; :: thesis:

then ((k1 + 1) - n) + 1 <= j by INT_1:7;
then (k1 + 1) - n <= j1 by XREAL_1:19;
then A57: i < j1 by A52, XXREAL_0:2;
y2 = (t . (fsloc 0)) . j1 by A9, A35, A28, A24, A31, A29, A21, A30, A34, A44, A51, A56, Lm19;
hence y1 >= y2 by A39, A43, A47, A53, A57; :: thesis:

end;

suppose A58: j = (k1 + 1) - n ; :: thesis:

A59: i < (k1 - n) + 1 by A52;
then A60: i <= k1 - n by INT_1:7;
A61: y2 = (t . (fsloc 0)) . (k1 + 1) by A28, A24, A31, A29, A33, A12, A51, A58, Lm19;

hereby :: thesis:

reconsider kn = k1 - n as Element of NAT by A59, INT_1:3, INT_1:7;
A62: (t . (fsloc 0)) . kn = x1 by A18, A43, A60, Lm20, XXREAL_0:2;

per cases ;

suppose i = kn ; :: thesis:

hence y1 >= y2 by A18, A28, A24, A31, A29, A21, A12, A25, A43, A50, A52, A61, A62, Lm19; :: thesis:

end;

suppose i <> kn ; :: thesis:

then i < kn by A60, XXREAL_0:1;
then y1 >= x1 by A26, A39, A43, A42, A53, XXREAL_0:2;
hence y1 >= y2 by A18, A28, A24, A25, A37, A43, A51, A58, A60, XXREAL_0:2; :: thesis:

end;

suppose A63: i > (k1 + 1) - n ; :: thesis:

then ((k1 + 1) - n) + 1 <= i by INT_1:7;
then 0 < i1 by A22, XREAL_1:19;
then A64: 1 + 0 <= i1 by INT_1:7;
(k1 + 1) - n < j by A45, A63, XXREAL_0:2;
then A65: y2 = (t . (fsloc 0)) . j1 by A9, A35, A28, A24, A31, A29, A21, A30, A34, A44, A51, Lm19;
i <= k1 + 1 by A9, A35, A34, A44, A45, XXREAL_0:2;
then y1 = (t . (fsloc 0)) . i1 by A28, A24, A31, A29, A21, A30, A50, A63, Lm19;
hence y1 >= y2 by A39, A47, A49, A65, A64; :: thesis:

end;

suppose A66: i = (k1 + 1) - n ; :: thesis:

k1 < k1 + 1 by XREAL_1:29;
then A67: j1 < k1 + 1 by A47, XXREAL_0:2;
((k1 + 1) - n) - 1 < j1 by A45, A66, XREAL_1:9;
then A68: ex yy being Integer st
( yy = (t . (fsloc 0)) . j1 & yy <= n3 ) by A14, A19, A25, A67;
y2 = (t . (fsloc 0)) . j1 by A9, A35, A28, A24, A31, A29, A33, A30, A34, A44, A45, A51, A66, Lm19;
hence y1 >= y2 by A28, A24, A31, A29, A33, A30, A50, A66, A68, Lm19; :: thesis:

end;

then (IExec (((((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),Q,t)) . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))))),Q,(IExec (((((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),Q,t)))) . (fsloc 0) are_fiberwise_equipotent by A8, A35, A36;
hence t . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))))),Q,t)) . (fsloc 0) are_fiberwise_equipotent by A28, A24, A32, A38, CLASSES1:76; :: thesis:
let i, j be Nat; :: thesis:
assume that
A69: i >= 1 and
A70: j <= len (t . (fsloc 0)) and
A71: i < j ; :: thesis:
let x1, x2 be Integer; :: thesis:
assume that
A72: x1 = ((IExec ((Times ((intloc (0 + 1)),(((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))))),Q,t)) . (fsloc 0)) . i and
A73: x2 = ((IExec ((Times ((intloc (0 + 1)),(((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))))),Q,t)) . (fsloc 0)) . j ; :: thesis:
j <= len ((IExec (((((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),Q,t)) . (fsloc 0)) by A28, A24, A32, A70, RFINSEQ:3;
hence x1 >= x2 by A8, A35, A36, A40, A38, A69, A71, A72, A73; :: thesis:

end;

hence S₁[k + 1] ; :: thesis:

end;

A74: S₁[ 0 ]

proof

let t be State of SCM+FSA; :: thesis:
let Q be Instruction-Sequence of SCM+FSA; :: thesis:
assume that
A75: t . (intloc (0 + 1)) = 0 and
t . (intloc (0 + 1)) <= (len (t . (fsloc 0))) - 1 ; :: thesis:
(IExec ((Times ((intloc (0 + 1)),(((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))))),Q,t)) . (fsloc 0) = t . (fsloc 0) by A75, Lm11, SCM_HALT:67;
hence ( ex i, j being Nat st
( i >= 1 & j <= (len (t . (fsloc 0))) - (t . (intloc (0 + 1))) & i < j & ex x1, x2 being Integer st
( x1 = (t . (fsloc 0)) . i & x2 = (t . (fsloc 0)) . j & not x1 >= x2 ) ) or ( t . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))))),Q,t)) . (fsloc 0) are_fiberwise_equipotent & ( for i, j being Nat st i >= 1 & j <= len (t . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))))),Q,t)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))))),Q,t)) . (fsloc 0)) . j holds
x1 >= x2 ) ) ) by A75; :: thesis:

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A74, A7);
then A76: S₁[m] ;
then ( ( for i, j being Nat st i >= 1 & j <= (len (s . (fsloc 0))) - (s . (intloc (0 + 1))) & i < j holds
for x1, x2 being Integer st x1 = (s . (fsloc 0)) . i & x2 = (s . (fsloc 0)) . j holds
x1 >= x2 ) implies ( s . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))))),P,s)) . (fsloc 0) are_fiberwise_equipotent & ( for i, j being Nat st i >= 1 & j <= len (s . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))))),P,s)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))))),P,s)) . (fsloc 0)) . j holds
x1 >= x2 ) ) ) by A1;
hence s . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))))),P,s)) . (fsloc 0) are_fiberwise_equipotent by A1, XXREAL_0:2; :: thesis:
for i, j being Nat st i >= 1 & j <= (len (s . (fsloc 0))) - (s . (intloc (0 + 1))) & i < j holds
for x1, x2 being Integer st x1 = (s . (fsloc 0)) . i & x2 = (s . (fsloc 0)) . j holds
x1 >= x2 by A1, XXREAL_0:2;
hence for i, j being Nat st i >= 1 & j <= len (s . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))))),P,s)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((((((intloc (1 + 1)) :=len (fsloc 0)) ";" (SubFrom ((intloc (1 + 1)),(intloc (0 + 1))))) ";" ((intloc (2 + 1)) := (intloc (1 + 1)))) ";" (AddTo ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (SubFrom ((intloc (3 + 1)),(intloc (3 + 1))))) ";" (while>0 ((intloc (1 + 1)),((((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1)))) ";" (SubFrom ((intloc (4 + 1)),(intloc (5 + 1))))) ";" (if>0 ((intloc (4 + 1)),(Macro (SubFrom ((intloc (1 + 1)),(intloc (1 + 1))))),((AddTo ((intloc (3 + 1)),(intloc 0))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))))))))) ";" (Times ((intloc (3 + 1)),(((((((intloc (1 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (1 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" (((fsloc 0),(intloc (1 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (4 + 1))))))))),P,s)) . (fsloc 0)) . j holds
x1 >= x2 by A1, A76; :: thesis:

end;