reserve L,L1,L2 for Lattice,
  F1,F2 for Filter of L,
  p,q,r,s for Element of L,
  p1,q1,r1,s1 for Element of L1,
  p2,q2,r2,s2 for Element of L2,
  X,x,x1,x2,y,y1,y2 for set,
  D,D1,D2 for non empty set,
  R for Relation,
  RD for Equivalence_Relation of D,
  a,b,d for Element of D,
  a1,b1,c1 for Element of D1,
  a2,b2,c2 for Element of D2,
  B for B_Lattice,
  FB for Filter of B,
  I for I_Lattice,
  FI for Filter of I ,
  i,i1,i2,j,j1,j2,k for Element of I,
  f1,g1 for BinOp of D1,
  f2,g2 for BinOp of D2;
reserve F,G for BinOp of D,RD;

theorem
  L1 is modular & L2 is modular iff [:L1,L2:] is modular
proof
  thus L1 is modular & L2 is modular implies [:L1,L2:] is modular
  proof
    assume
A1: for p1,q1,r1 st p1 [= r1 holds p1"\/"(q1"/\"r1) = (p1"\/"q1)"/\"r1;
    assume
A2: for p2,q2,r2 st p2 [= r2 holds p2"\/"(q2"/\"r2) = (p2"\/"q2)"/\"r2;
    let a,b,c be Element of [:L1,L2:] such that
A3: a [= c;
    consider q1,q2 such that
A4: b = [q1,q2] by DOMAIN_1:1;
    consider p1,p2 such that
A5: a = [p1,p2] by DOMAIN_1:1;
    consider r1,r2 such that
A6: c = [r1,r2] by DOMAIN_1:1;
A7: p2 [= r2 by A3,A5,A6,Th36;
A8: p1 [= r1 by A3,A5,A6,Th36;
    thus a"\/"(b"/\"c) = a"\/"([q1"/\"r1,q2"/\"r2]) by A4,A6,Th21
      .= [p1"\/"(q1"/\"r1),p2"\/"(q2"/\"r2)] by A5,Th21
      .= [(p1"\/"q1)"/\"r1,p2"\/"(q2"/\"r2)] by A1,A8
      .= [(p1"\/"q1)"/\"r1,(p2"\/"q2)"/\"r2] by A2,A7
      .= [p1"\/"q1,p2"\/"q2]"/\"c by A6,Th21
      .= (a"\/"b)"/\"c by A5,A4,Th21;
  end;
  assume
A9: for a,b,c be Element of [:L1,L2:] st a [= c holds a"\/"(b"/\"c) = (a
  "\/"b)"/\"c;
  thus L1 is modular
  proof
    set p2 = the Element of L2;
    let p1,q1,r1;
    assume p1 [= r1;
    then [p1,p2] [= [r1,p2] by Th36;
    then
A10: [p1,p2]"\/"([q1,p2]"/\"[r1,p2]) = ([p1,p2]"\/"[q1,p2])"/\"[r1,p2] by A9;
A11: [p1,p2]"\/"[q1,p2] = [p1"\/"q1,p2"\/"p2] by Th21;
A12: [p1"\/"q1,p2"\/"p2]"/\"[r1,p2] = [(p1"\/"q1)"/\"r1,(p2"\/"p2)"/\" p2]
    by Th21;
A13: [p1,p2]"\/"[q1"/\"r1,p2"/\"p2] = [p1"\/"(q1"/\"r1),p2"\/"(p2"/\"p2)]
    by Th21;
    [q1,p2]"/\"[r1,p2] = [q1"/\"r1,p2"/\"p2] by Th21;
    hence thesis by A10,A11,A13,A12,XTUPLE_0:1;
  end;
  set p1 = the Element of L1;
  let p2,q2,r2;
  assume p2 [= r2;
  then [p1,p2] [= [p1,r2] by Th36;
  then
A14: [p1,p2]"\/"([p1,q2]"/\"[p1,r2]) = ([p1,p2]"\/"[p1,q2])"/\"[p1,r2] by A9;
A15: [p1,p2]"\/"[p1,q2] = [p1"\/"p1,p2"\/"q2] by Th21;
A16: [p1"\/"p1,p2"\/"q2]"/\"[p1,r2] = [(p1"\/"p1)"/\"p1,(p2"\/"q2)"/\" r2]
  by Th21;
A17: [p1,p2]"\/"[p1"/\"p1,q2"/\"r2] = [p1"\/"(p1"/\"p1),p2"\/"(q2"/\"r2)] by
Th21;
  [p1,q2]"/\"[p1,r2] = [p1"/\"p1,q2"/\"r2] by Th21;
  hence thesis by A14,A15,A17,A16,XTUPLE_0:1;
end;
