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 Th44:
  A<>B & L1 = Line(A,B) & L1 _|_ L2 & the_midpoint_of_the_segment(A,B) in L2
  implies L2 = the_perpendicular_bisector(A,B)
  proof
    assume that
A1: A<>B and
A2: L1 = Line(A,B) and
A3: L1 _|_ L2 and
A4: the_midpoint_of_the_segment(A,B) in L2;
    set M = the_midpoint_of_the_segment(A,B);
    consider L3, L4 be Element of line_of_REAL 2 such that
A5: the_perpendicular_bisector(A,B) = L4 and
A6: L3 = Line(A,B) and
A7: L3 _|_ L4 and
A8: L3 /\ L4 = {the_midpoint_of_the_segment(A,B)} by A1,Def2;
A9: L2 // L4 by A2,A3,A6,A7,Th13,EUCLIDLP:111;
    M in L3/\L4 by A8,TARSKI:def 1;
    then M in L2 & M in L4 by A4,XBOOLE_0:def 4;
    hence thesis by A5,A9,XBOOLE_0:3,EUCLIDLP:71;
  end;
