let A be preIfWhileAlgebra; :: thesis: for I being Element of A
for S being non empty set
for T being Subset of S
for s being Element of S
for f being ExecutionFunction of A,S,T holds iteration-degree I,s,f >= 0
let I be Element of A; :: thesis: for S being non empty set
for T being Subset of S
for s being Element of S
for f being ExecutionFunction of A,S,T holds iteration-degree I,s,f >= 0
let S be non empty set ; :: thesis: for T being Subset of S
for s being Element of S
for f being ExecutionFunction of A,S,T holds iteration-degree I,s,f >= 0
let T be Subset of S; :: thesis: for s being Element of S
for f being ExecutionFunction of A,S,T holds iteration-degree I,s,f >= 0
let s be Element of S; :: thesis: for f being ExecutionFunction of A,S,T holds iteration-degree I,s,f >= 0
let f be ExecutionFunction of A,S,T; :: thesis: iteration-degree I,s,f >= 0

per cases ( not f iteration_terminates_for I,s or f iteration_terminates_for I,s ) ;

suppose not f iteration_terminates_for I,s ; :: thesis: iteration-degree I,s,f >= 0

hence iteration-degree I,s,f >= 0 by Def34; :: thesis: verum

end;

suppose f iteration_terminates_for I,s ; :: thesis: iteration-degree I,s,f >= 0

then consider r being non empty FinSequence of S such that
A1: iteration-degree I,s,f = (len r) - 1 and
( r . 1 = s & r . (len r) nin T & ( for i being Nat st 1 <= i & i < len r holds
( r . i in T & r . (i + 1) = f . (r . i),I ) ) ) by Def34;
ex i being Nat st len r = i + 1 by NAT_1:6;
hence iteration-degree I,s,f >= 0 by A1; :: thesis: verum

end;

end;