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

theorem Th25:
  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.];
    x in [.x,y.] by A1,XXREAL_1:1;
    hence thesis by Def2;
  end;
  x is LowerBound of [.x,y.] by Th17;
  hence thesis by A2,Def4;
end;
