theorem Th41:
  A <> B & C in the_perpendicular_bisector(A,B) implies |.C-A.| = |.C-B.|
  proof
    assume that
A1: A<>B and
A2: C in the_perpendicular_bisector(A,B);
    consider L1, L2 be Element of line_of_REAL 2 such that
A3: the_perpendicular_bisector(A,B) = L2 and
A4: L1 = Line(A,B) and
A5: L1 _|_ L2 and
A6: L1 /\ L2 = {the_midpoint_of_the_segment(A,B)} by A1,Def2;
    set D = the_midpoint_of_the_segment(A,B);
    now
      thus A<>B by A1;
      thus L1 = Line(A,B) by A4;
      thus D in LSeg(A,B) by Th21;
      consider E such that
      E in LSeg(A,B) and
A7:   the_midpoint_of_the_segment(A,B) = E and
A8:   |.A-E.| = half_length(A,B) by Def1;
      thus |.A-D.|=1/2*|.A-B.| by A7,A8;
      D in L1/\L2 by A6,TARSKI:def 1;
      hence D in L2 by XBOOLE_0:def 4;
      thus L1 _|_ L2 by A5;
      thus C in L2 by A2,A3;
    end;
    hence thesis by Th40;
  end;
