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

theorem Th7:
  for x being LowerBound of A, y being LowerBound of B holds min(x,
  y) is LowerBound of A\/ B
proof
  let x be LowerBound of A, y be LowerBound of B;
  set m = min(x,y);
  let z;
  assume
A1: z in A \/ B;
  per cases by A1,XBOOLE_0:def 3;
  suppose
A2: z in A;
A3: m <= x by XXREAL_0:17;
    x <= z by A2,Def2;
    hence thesis by A3,XXREAL_0:2;
  end;
  suppose
A4: z in B;
A5: m <= y by XXREAL_0:17;
    y <= z by A4,Def2;
    hence thesis by A5,XXREAL_0:2;
  end;
end;
