for L being lower-bounded modular LATTICE
for S being ExtensionSeq2 of the carrier of L, BasicDF L
for FS being non empty set
for FD being distance_function of FS,L
for f being Homomorphism of L,(EqRelLATT FS)
for e1, e2 being Equivalence_Relation of FS
for x, y being object st f = alpha FD & FS = union { ((S . i) `1) where i is Element of NAT : verum } & FD = union { ((S . i) `2) where i is Element of NAT : verum } & e1 in the carrier of (Image f) & e2 in the carrier of (Image f) & [x,y] in e1 "\/" e2 holds
ex F being non empty FinSequence of FS st
( len F = 2 + 2 & x,y are_joint_by F,e1,e2 )