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 Th46:
  A,B,C is_a_triangle implies
  the_perpendicular_bisector(A,B) /\ the_perpendicular_bisector(B,C)
  is being_point
  proof
    assume
A1: A,B,C is_a_triangle;
    then
A2: A,B,C are_mutually_distinct by EUCLID_6:20;
    set MAB = the_perpendicular_bisector(A,B);
    set MiAB = the_midpoint_of_the_segment(A,B);
    set MBC = the_perpendicular_bisector(B,C);
    set MiBC = the_midpoint_of_the_segment(B,C);
    consider LAB1,LAB2 be Element of line_of_REAL 2 such that
A3: MAB = LAB2 and
A4: LAB1 = Line(A,B) and
A5: LAB1 _|_ LAB2 and
    LAB1 /\ LAB2 = {MiAB} by A2,Def2;
    consider LBC1,LBC2 be Element of line_of_REAL 2 such that
A6: MBC = LBC2 and
A7: LBC1 = Line(B,C) and
A8: LBC1 _|_ LBC2 and
    LBC1 /\ LBC2 = {MiBC} by A2,Def2;
    now
      hereby
        assume LAB2 // LBC2;
        then LBC1 _|_ LAB2 by A8,EUCLIDLP:61;
        then
A9:     LAB1 // LBC1 by A5,Th13,EUCLIDLP:111;
        B in LAB1 & B in LBC1 by A4,A7,RLTOPSP1:72;
        then LAB1=LBC1 by A9,EUCLIDLP:71,XBOOLE_0:3;
        then C in LAB1 by A7,RLTOPSP1:72;
        then C,A,B are_collinear by A2,A4,MENELAUS:13;
        hence contradiction by A1,MENELAUS:15;
      end;
      thus LAB2 is being_line by A5,EUCLIDLP:67;
      thus LBC2 is being_line by A8,EUCLIDLP:67;
    end;
    then not LAB2 // LBC2 & not LAB2 is being_point & not LBC2 is being_point
      by Th7;
    hence thesis by A3,A6,Th16;
  end;
