theorem
  A <> B implies the_perpendicular_bisector(A,B) is being_line
  proof
    assume A<>B;
    then consider L1, L2 be Element of line_of_REAL 2 such that
A1: the_perpendicular_bisector(A,B) = L2 and
    L1=Line(A,B) and
A2: L1_|_L2 and
    L1/\L2= {the_midpoint_of_the_segment(A,B)} by Def2;
    thus thesis by A1,A2,EUCLIDLP:67;
end;
