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) = the_perpendicular_bisector(B,A)
  proof
    assume
A1: A<>B;
    then consider L1, L2 be Element of line_of_REAL 2 such that
A2: the_perpendicular_bisector(A,B) = L2 and
A3: L1 = Line(A,B) and
A4: L1 _|_ L2 and
A5: L1 /\ L2= {the_midpoint_of_the_segment(A,B)} by Def2;
    consider L3, L4 be Element of line_of_REAL 2 such that
A6: the_perpendicular_bisector(B,A) = L4 and
A7: L3 = Line(A,B) and
A8: L3 _|_ L4 and
A9: L3 /\ L4 = {the_midpoint_of_the_segment(B,A)} by A1,Def2;
    set M=the_midpoint_of_the_segment(A,B);
    consider x such that
A10: L1 /\ L2 = {x} by A4,Th33;
    consider y such that
A11: L3 /\ L4= {y} by A8,Th33;
A12: L2 // L4 by A3,A4,A7,A8,Th13,EUCLIDLP:111;
    {x}={M} & {y}={M} by A5,A9,A10,A11,Th23;
    then M in L1/\L2 & M in L3/\L4 by A10,A11,TARSKI:def 1;
    then M in L1 & M in L2 & M in L3 & M in L4 by XBOOLE_0:def 4;
    hence thesis by A2,A6,A12,XBOOLE_0:3,EUCLIDLP:71;
  end;
