let X be non empty set ; :: thesis: for R, S being RMembership_Func of X,X
for n being natural number st S c= holds
n iter S c=
let R, S be RMembership_Func of X,X; :: thesis: for n being natural number st S c= holds
n iter S c=
let n be natural number ; :: thesis: ( S c= implies n iter S c= )
assume A1: S c= ; :: thesis: n iter S c=
defpred S₁[ natural number ] means $1 iter S c= ;
0 iter R = Imf X,X by Th24
.= 0 iter S by Th24 ;
then A2: S₁[ 0 ] ;
A3: for k being natural number st S₁[k] holds
S₁[k + 1]

proof

let k be natural number ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A4: S₁[k] ; :: thesis: S₁[k + 1]
( (k iter R) (#) R = (k + 1) iter R & (k iter S) (#) S = (k + 1) iter S ) by Th26;
hence S₁[k + 1] by A1, A4, Th6; :: thesis: verum

end;

for k being natural number holds S₁[k] from NAT_1:sch 2(A2, A3);
hence n iter S c= ; :: thesis: verum