reserve x,y,z,r,s for ExtReal;
reserve A,B for ext-real-membered set;

theorem Th9:
  inf(A \/ B) = min(inf A,inf B)
proof
  set m = min(inf A,inf B);
A1: inf B is LowerBound of B by Def4;
A2: for x being LowerBound of A \/ B holds x <= m
  proof
    let x be LowerBound of A \/ B;
    x is LowerBound of B by Th5,XBOOLE_1:7;
    then
A3: x <= inf B by Def4;
    x is LowerBound of A by Th5,XBOOLE_1:7;
    then x <= inf A by Def4;
    hence thesis by A3,XXREAL_0:20;
  end;
  inf A is LowerBound of A by Def4;
  then m is LowerBound of A \/ B by A1,Th7;
  hence thesis by A2,Def4;
end;
