
theorem
  for L1,L2 being antisymmetric non empty RelStr for D being non empty
Subset of [:L1,L2:] st [:L1,L2:] is complete or ex_inf_of D,[:L1,L2:] holds inf
  D = [inf proj1 D,inf proj2 D]
proof
  let L1,L2 be antisymmetric non empty RelStr, D be non empty Subset of [:L1,
  L2:];
  reconsider C1 = the carrier of L1, C2 = the carrier of L2 as non empty set;
  the carrier of [:L1,L2:] = [:C1,C2:] by Def2;
  then consider d1, d2 being object such that
A1: d1 in C1 and
A2: d2 in C2 and
A3: inf D = [d1,d2] by ZFMISC_1:def 2;
  reconsider d1 as Element of L1 by A1;
  reconsider D9 = D as non empty Subset of [:C1,C2:] by Def2;
  proj1 D9 is non empty;
  then reconsider D1 = proj1 D as non empty Subset of L1;
  proj2 D9 is non empty;
  then reconsider D2 = proj2 D as non empty Subset of L2;
A4: D9 c= [:D1,D2:] by Th1;
  reconsider d2 as Element of L2 by A2;
  assume [:L1,L2:] is complete or ex_inf_of D,[:L1,L2:];
  then
A5: ex_inf_of D,[:L1,L2:] by YELLOW_0:17;
  then
A6: ex_inf_of D2,L2 by Th42;
A7: ex_inf_of D1,L1 by A5,Th42;
  then ex_inf_of [:D1,D2:],[:L1,L2:] by A6,Th40;
  then inf D >= inf [:D1,D2:] by A5,A4,YELLOW_0:35;
  then
A8: inf D >= [inf proj1 D,inf proj2 D] by A7,A6,Th44;
  D2 is_>=_than d2
  proof
    let b be Element of L2;
    assume b in D2;
    then consider x being object such that
A9: [x,b] in D by XTUPLE_0:def 13;
    reconsider x as Element of D1 by A9,XTUPLE_0:def 12;
    reconsider x as Element of L1;
    D is_>=_than [d1,d2] by A5,A3,YELLOW_0:def 10;
    then [x,b] >= [d1,d2] by A9;
    hence thesis by Th11;
  end;
  then
A10: inf D2 >= d2 by A6,YELLOW_0:def 10;
  D1 is_>=_than d1
  proof
    let b be Element of L1;
    assume b in D1;
    then consider x being object such that
A11: [b,x] in D by XTUPLE_0:def 12;
    reconsider x as Element of D2 by A11,XTUPLE_0:def 13;
    reconsider x as Element of L2;
    D is_>=_than [d1,d2] by A5,A3,YELLOW_0:def 10;
    then [b,x] >= [d1,d2] by A11;
    hence thesis by Th11;
  end;
  then inf D1 >= d1 by A7,YELLOW_0:def 10;
  then [inf D1,inf D2] >= inf D by A3,A10,Th11;
  hence thesis by A8,ORDERS_2:2;
end;
