
theorem Th7: :: generealized YELLOW_3:47, YELLOW10:6
  for L1,L2 being antisymmetric non empty RelStr, D being Subset
  of [:L1,L2:] st 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 Subset of [:L1,L2:] such
  that
A1: ex_inf_of D,[:L1,L2:];
  per cases;
  suppose
    D <> {};
    hence thesis by A1,YELLOW_3:47;
  end;
  suppose
A2: D = {};
    then ex_inf_of {},L2 by A1,FUNCT_5:8,YELLOW_3:42;
    then
A3: L2 is upper-bounded by Th5;
    ex_inf_of {},L1 by A1,A2,FUNCT_5:8,YELLOW_3:42;
    then L1 is upper-bounded by Th5;
    hence thesis by A1,A3,YELLOW10:6;
  end;
end;
