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 Th22:
  the_midpoint_of_the_segment(A,B) = 1/2 * (A+B)
  proof
    now
      1/2 * (A+B) = (1-1/2) * A + 1/2 * B by RLVECT_1:def 5;
      then 1/2 * (A+B) in {(1-r)*A+r*B:0<=r<=1};
      hence 1/2 * (A+B) in LSeg(A,B) by RLTOPSP1:def 2;
      thus |.A-(1/2 *(A+B)).| = half_length(A,B) by Th18;
    end;
    hence thesis by Def1;
  end;
