let X1, X2, X3, X4, Y1, Y2, Y3, Y4 be set ; :: thesis: ( [:X1,X2,X3,X4:] <> {} & [:X1,X2,X3,X4:] = [:Y1,Y2,Y3,Y4:] implies ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 ) )
assume [:X1,X2,X3,X4:] <> {} ; :: thesis: ( not [:X1,X2,X3,X4:] = [:Y1,Y2,Y3,Y4:] or ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 ) )
then ( X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} ) by Th55;
hence ( not [:X1,X2,X3,X4:] = [:Y1,Y2,Y3,Y4:] or ( X1 = Y1 & X2 = Y2 & X3 = Y3 & X4 = Y4 ) ) by Th56; :: thesis: verum