theorem
  latt (L,(.L.>) = the LattStr of L & latt (L,<.L.)) = the LattStr of L
proof
A1: dom met(L) = [:carr(L), carr(L):] & join(L)|(dom join(L) qua set) = join
  (L) by FUNCT_2:def 1,RELAT_1:68;
  (ex o1,o2 being BinOp of (.L.> st o1 = (the L_join of L)|| (.L.> & o2 =
  (the L_meet of L)||(.L.> & latt (L,(.L.>) = LattStr (#(.L.>, o1, o2#) )& dom
  join( L) = [:carr(L), carr(L):] by Def14,FUNCT_2:def 1;
  hence thesis by A1,RELAT_1:68;
end;
