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 implicative & L2 is implicative iff [:L1,L2:] is implicative
proof
  thus L1 is implicative & L2 is implicative implies [:L1,L2:] is implicative
  proof
    assume
A1: for p1,q1 ex r1 st p1"/\"r1 [= q1 & for s1 st p1"/\"s1 [= q1 holds
    s1 [= r1;
    assume
A2: for p2,q2 ex r2 st p2"/\"r2 [= q2 & for s2 st p2"/\"s2 [= q2 holds
    s2 [= r2;
    let a,b be Element of [:L1,L2:];
    consider p1,p2 such that
A3: a = [p1,p2] by DOMAIN_1:1;
    consider q1,q2 such that
A4: b = [q1,q2] by DOMAIN_1:1;
    consider r2 such that
A5: p2"/\"r2 [= q2 and
A6: for s2 st p2"/\"s2 [= q2 holds s2 [= r2 by A2;
    consider r1 such that
A7: p1"/\"r1 [= q1 and
A8: for s1 st p1"/\"s1 [= q1 holds s1 [= r1 by A1;
    take [r1,r2];
    a"/\"[r1,r2] = [p1"/\"r1,p2"/\"r2] by A3,Th21;
    hence a"/\"[r1,r2] [= b by A4,A7,A5,Th36;
    let d be Element of [:L1,L2:];
    consider s1,s2 such that
A9: d = [s1,s2] by DOMAIN_1:1;
    assume a"/\"d [= b;
    then
A10: [p1"/\"s1,p2"/\"s2] [= b by A3,A9,Th21;
    then p2"/\"s2 [= q2 by A4,Th36;
    then
A11: s2 [= r2 by A6;
    p1"/\"s1 [= q1 by A4,A10,Th36;
    then s1 [= r1 by A8;
    hence d [= [r1,r2] by A9,A11,Th36;
  end;
  assume
A12: for a,b being Element of [:L1,L2:] ex c being Element of [:L1,L2:]
st a"/\"c [= b & for d being Element of [:L1,L2:] st a"/\" d [= b holds d [= c;
  thus for p1,q1 ex r1 st p1"/\"r1 [= q1 & for s1 st p1"/\"s1 [= q1 holds s1
  [= r1
  proof
    set p2 = the Element of L2;
    let p1,q1;
    consider c being Element of [:L1,L2:] such that
A13: [p1,p2]"/\"c [= [q1,p2] and
A14: for d being Element of [:L1,L2:] st [p1,p2]"/\"d [= [q1,p2] holds
    d [= c by A12;
    consider r1,r2 such that
A15: c = [r1,r2] by DOMAIN_1:1;
    take r1;
A16: [p1,p2]"/\"c = [p1"/\"r1,p2"/\"r2] by A15,Th21;
    hence p1"/\"r1 [= q1 by A13,Th36;
    let s1;
    assume
A17: p1"/\"s1 [= q1;
    p2"/\"r2 [= p2 by A13,A16,Th36;
    then [p1"/\"s1,p2"/\"r2] [= [q1,p2] by A17,Th36;
    then [p1,p2]"/\"[s1,r2] [= [q1,p2] by Th21;
    then [s1,r2] [= c by A14;
    hence thesis by A15,Th36;
  end;
  set p1 = the Element of L1;
  let p2,q2;
  consider c being Element of [:L1,L2:] such that
A18: [p1,p2]"/\"c [= [p1,q2] and
A19: for d being Element of [:L1,L2:] st [p1,p2]"/\"d [= [p1,q2] holds d
  [= c by A12;
  consider r1,r2 such that
A20: c = [r1,r2] by DOMAIN_1:1;
  take r2;
A21: [p1,p2]"/\"c = [p1"/\"r1,p2"/\"r2] by A20,Th21;
  hence p2"/\"r2 [= q2 by A18,Th36;
  let s2;
  assume
A22: p2"/\"s2 [= q2;
  p1"/\"r1 [= p1 by A18,A21,Th36;
  then [p1"/\"r1,p2"/\"s2] [= [p1,q2] by A22,Th36;
  then [p1,p2]"/\"[r1,s2] [= [p1,q2] by Th21;
  then [r1,s2] [= c by A19;
  hence s2 [= r2 by A20,Th36;
end;
