let I be set ; :: thesis: for X, Y being ManySortedSet of I holds
( X \/ (X /\ Y) = X & (X /\ Y) \/ X = X & X \/ (Y /\ X) = X & (Y /\ X) \/ X = X )
let X, Y be ManySortedSet of I; :: thesis: ( X \/ (X /\ Y) = X & (X /\ Y) \/ X = X & X \/ (Y /\ X) = X & (Y /\ X) \/ X = X )
X /\ Y c= X by Th17;
then A1: X \/ (X /\ Y) c= X by Th18;
A2: X c= X \/ (X /\ Y) by Th16;
hence X \/ (X /\ Y) = X by A1, Def13; :: thesis: ( (X /\ Y) \/ X = X & X \/ (Y /\ X) = X & (Y /\ X) \/ X = X )
thus (X /\ Y) \/ X = X by A1, A2, Def13; :: thesis: ( X \/ (Y /\ X) = X & (Y /\ X) \/ X = X )
thus ( X \/ (Y /\ X) = X & (Y /\ X) \/ X = X ) by A1, A2, Def13; :: thesis: verum