let it1, it2 be Nat; :: thesis: ( ex k being Nat st
( it1 = k & ((StepWhile>0 (a,I,p,s)) . k) . a <= 0 & ( for i being Nat st ((StepWhile>0 (a,I,p,s)) . i) . a <= 0 holds
k <= i ) & DataPart (Comput ((p +* (while>0 (a,I))),(Initialize s),(LifeSpan ((p +* (while>0 (a,I))),(Initialize s))))) = DataPart ((StepWhile>0 (a,I,p,s)) . k) ) & ex k being Nat st
( it2 = k & ((StepWhile>0 (a,I,p,s)) . k) . a <= 0 & ( for i being Nat st ((StepWhile>0 (a,I,p,s)) . i) . a <= 0 holds
k <= i ) & DataPart (Comput ((p +* (while>0 (a,I))),(Initialize s),(LifeSpan ((p +* (while>0 (a,I))),(Initialize s))))) = DataPart ((StepWhile>0 (a,I,p,s)) . k) ) implies it1 = it2 )
given k1 being Nat such that A31: it1 = k1 and
A32: ( ((StepWhile>0 (a,I,p,s)) . k1) . a <= 0 & ( for i being Nat st ((StepWhile>0 (a,I,p,s)) . i) . a <= 0 holds
k1 <= i ) ) and
DataPart (Comput ((p +* (while>0 (a,I))),(Initialize s),(LifeSpan ((p +* (while>0 (a,I))),(Initialize s))))) = DataPart ((StepWhile>0 (a,I,p,s)) . k1) ; :: thesis: ( for k being Nat holds
( not it2 = k or not ((StepWhile>0 (a,I,p,s)) . k) . a <= 0 or ex i being Nat st
( ((StepWhile>0 (a,I,p,s)) . i) . a <= 0 & not k <= i ) or not DataPart (Comput ((p +* (while>0 (a,I))),(Initialize s),(LifeSpan ((p +* (while>0 (a,I))),(Initialize s))))) = DataPart ((StepWhile>0 (a,I,p,s)) . k) ) or it1 = it2 )
given k2 being Nat such that A33: it2 = k2 and
A34: ( ((StepWhile>0 (a,I,p,s)) . k2) . a <= 0 & ( for i being Nat st ((StepWhile>0 (a,I,p,s)) . i) . a <= 0 holds
k2 <= i ) ) and
DataPart (Comput ((p +* (while>0 (a,I))),(Initialize s),(LifeSpan ((p +* (while>0 (a,I))),(Initialize s))))) = DataPart ((StepWhile>0 (a,I,p,s)) . k2) ; :: thesis: it1 = it2
( k1 <= k2 & k2 <= k1 ) by A32, A34;
hence it1 = it2 by A31, A33, XXREAL_0:1; :: thesis: verum