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;
