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;
