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

theorem
  min{x,y} = min(x,y)
proof
  now
    per cases;
    case
A1:   y <= x;
A2:   for z being LowerBound of {x,y} holds z <= y by Lm6;
      y is LowerBound of {x,y} by A1,Lm5;
      hence min{x,y} = y by A2,Def4;
    end;
    case
A3:   x < y;
A4:   for z being LowerBound of {x,y} holds z <= x by Lm6;
      x is LowerBound of {x,y} by A3,Lm5;
      hence min{x,y} = x by A4,Def4;
    end;
  end;
  hence thesis by XXREAL_0:def 9;
end;
