
theorem Th5:
  for L1,L2 be lower-bounded LATTICE st L1,L2 are_isomorphic & L1
  is modular holds L2 is modular
proof
  let L1,L2 be lower-bounded LATTICE;
  assume that
A1: L1,L2 are_isomorphic and
A2: L1 is modular;
  let a,b,c be Element of L2;
  consider f be Function of L1,L2 such that
A3: f is isomorphic by A1,WAYBEL_1:def 8;
  set C = f".c;
  set A = f".a;
  set B = f".b;
A4: f is one-to-one & rng f = the carrier of L2 by A3,WAYBEL_0:66;
  then
A5: b = f.B by FUNCT_1:35;
A6: C in dom f by A4,FUNCT_1:32;
A7: A in dom f & B in dom f by A4,FUNCT_1:32;
A8: a = f.A & c = f.C by A4,FUNCT_1:35;
  reconsider A,B,C as Element of L1 by A7,A6;
  assume a <= c;
  then A <= C by A3,A8,WAYBEL_0:66;
  then
A9: A"\/"(B"/\"C) = (A"\/"B)"/\"C by A2;
  f is infs-preserving sups-preserving by A3,WAYBEL13:20;
  then
A10: f is meet-preserving join-preserving;
  hence a"\/"(b"/\"c) = f.A "\/" f.(B"/\"C) by A5,A8,WAYBEL_6:1
    .= f.((A"\/"B)"/\"C) by A10,A9,WAYBEL_6:2
    .= (f.(A"\/"B)"/\"f.C) by A10,WAYBEL_6:1
    .= (a"\/"b)"/\"c by A10,A5,A8,WAYBEL_6:2;
end;
