theorem Th26:
  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_1:3;
    hence thesis by Def2;
  end;
  x is LowerBound of [.x,y.[ by Th19;
  hence thesis by A2,Def4;
end;
