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

theorem Th27:
  x < y implies inf ].x,y.] = x
proof
  assume
A1: x < y;
A2: for z being LowerBound of ].x,y.] holds z <= x
  proof
    let z be LowerBound of ].x,y.];
    for r st x < r & r < y holds z <= r
    proof
      let r;
      assume that
A3:   x < r and
A4:   r < y;
      r in ].x,y.] by A3,A4,XXREAL_1:2;
      hence thesis by Def2;
    end;
    hence thesis by A1,XREAL_1:228;
  end;
  x is LowerBound of ].x,y.] by Th18;
  hence thesis by A2,Def4;
end;
