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 Th27:
  C in LSeg(A,B) & |.A-C.| = |.B-C.| implies half_length(A,B) = |.A-C.|
  proof
    assume that
A1: C in LSeg(A,B) and
A2: |.A-C.| = |.B-C.|;
A3: |.C-B.| = |.A-C.| by A2,EUCLID_6:43;
    half_length(A,B) = 1/2* (|.A-C.|+|.C-B.|) by A1,Th8
                    .= 1/2* (2*|.A-C.|) by A3;
    hence thesis;
  end;
