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 Th43:
  A <> B implies
  the_midpoint_of_the_segment(A,B) in the_perpendicular_bisector(A,B)
  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) & L1 _|_ L2 and
A2: L1 /\ L2 = {the_midpoint_of_the_segment(A,B)} by Def2;
    the_midpoint_of_the_segment(A,B) in L1/\L2 by A2,TARSKI:def 1;
    hence thesis by A1,XBOOLE_0:def 4;
  end;
