let it1, it2 be Element of NAT ; :: thesis: ( ex k being Element of NAT st
( it1 = k & ((StepWhile>0 a,I,s) . k) . a <= 0 & ( for i being Element of NAT st ((StepWhile>0 a,I,s) . i) . a <= 0 holds
k <= i ) & 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) . k) ) & ex k being Element of NAT st
( it2 = k & ((StepWhile>0 a,I,s) . k) . a <= 0 & ( for i being Element of NAT st ((StepWhile>0 a,I,s) . i) . a <= 0 holds
k <= i ) & 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) . k) ) implies it1 = it2 )
given k1 being Element of NAT such that A34: it1 = k1 and
A35: ((StepWhile>0 a,I,s) . k1) . a <= 0 and
A36: for i being Element of NAT st ((StepWhile>0 a,I,s) . i) . a <= 0 holds
k1 <= i and
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) . k1) ; :: thesis: ( for k being Element of NAT holds
( not it2 = k or not ((StepWhile>0 a,I,s) . k) . a <= 0 or ex i being Element of NAT st
( ((StepWhile>0 a,I,s) . i) . a <= 0 & not k <= i ) or not 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) . k) ) or it1 = it2 )
given k2 being Element of NAT such that A37: it2 = k2 and
A38: ((StepWhile>0 a,I,s) . k2) . a <= 0 and
A39: for i being Element of NAT st ((StepWhile>0 a,I,s) . i) . a <= 0 holds
k2 <= i and
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) . k2) ; :: thesis: it1 = it2
( k1 <= k2 & k2 <= k1 ) by A35, A36, A38, A39;
hence it1 = it2 by A34, A37, XXREAL_0:1; :: thesis: verum