reserve n for Nat,
        lambda,lambda2,mu,mu2 for Real,
        x1,x2 for Element of REAL n,
        An,Bn,Cn for Point of TOP-REAL n,
        a for Real;
 reserve Pn,PAn,PBn for Element of REAL n,
         Ln for Element of line_of_REAL n;
reserve A,B,C for Point of TOP-REAL 2;
reserve x,y,z,y1,y2 for Element of REAL 2;
reserve L,L1,L2,L3,L4 for Element of line_of_REAL 2;
reserve D,E,F for Point of TOP-REAL 2;
reserve b,c,d,r,s for Real;

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;
