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 Th39:
  A <> B & L1 = Line(A,B) & C in LSeg(A,B) & |.A-C.| = 1/2 * |.A-B.|
  implies ex L2 st C in L2 & L1 _|_ L2
  proof
    assume
A1: A <> B & L1 = Line(A,B) & C in LSeg(A,B) & |.A-C.| = 1/2 * |.A-B.|;
    reconsider x1=A,x2=B,x3=C as Element of REAL 2 by EUCLID:22;
    Line(A,B) = Line(x1,x2) by Th4;
    then L1 is being_line by A1;
    then consider L2 such that
A2: x3 in L2 and
A3: L1 _|_ L2 by Th35;
    thus thesis by A2,A3;
  end;
