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 Th47:
  A,B,C is_a_triangle implies ex D st
  the_perpendicular_bisector(A,B) /\ the_perpendicular_bisector(B,C) = {D} &
  the_perpendicular_bisector(B,C) /\ the_perpendicular_bisector(C,A) = {D} &
  the_perpendicular_bisector(C,A) /\ the_perpendicular_bisector(A,B) = {D} &
  |.D-A.| = |.D-B.| & |.D-A.| = |.D-C.| & |.D-B.| = |.D-C.|
  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);
    set MCA = the_perpendicular_bisector(C,A);
    set MiCA = the_midpoint_of_the_segment(C,A);
    MAB /\ MBC is being_point by A1,Th46;
    then consider x such that
A3: MAB /\ MBC ={x};
    B,C,A is_a_triangle by A1,MENELAUS:15;
    then MBC /\ MCA is being_point by Th46;
    then consider y such that
A4: MBC /\ MCA ={y};
    C,A,B is_a_triangle by A1,MENELAUS:15;
    then MCA /\ MAB is being_point by Th46;
    then consider z such that
A5: MCA /\ MAB ={z};
    x in MAB/\MBC by A3,TARSKI:def 1;
    then
A6: x in MAB & x in MBC by XBOOLE_0:def 4;
    reconsider Px=x as Point of TOP-REAL 2 by EUCLID:22;
A7: |.Px-A.| = |.Px-B.| & |.Px-B.|=|.Px-C.| by A6,Th41;
    y in MBC/\MCA by A4,TARSKI:def 1;
    then
A8: y in MBC & y in MCA by XBOOLE_0:def 4;
    reconsider Py=y as Point of TOP-REAL 2 by EUCLID:22;
    |.Py-B.| = |.Py-C.| & |.Py-C.| = |.Py-A.| by A8,Th41;
    then Py in MAB by A2,Th45;
    then Py in MAB/\MBC by A8,XBOOLE_0:def 4;
    then
A9: Py = Px by A3,TARSKI:def 1;
    z in MCA/\MAB by A5,TARSKI:def 1;
    then
A10: z in MCA & z in MAB by XBOOLE_0:def 4;
    reconsider Pz=z as Point of TOP-REAL 2 by EUCLID:22;
    |.Pz-C.| = |.Pz-A.| & |.Pz-A.| = |.Pz-B.| by A10,Th41;
    then Pz in MBC by A2,Th45;
    then
A11: Pz in MBC/\MCA by A10,XBOOLE_0:def 4;
    take Px;
    thus thesis by A7,A3,A4,A5,TARSKI:def 1,A11,A9;
  end;
