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 Th25:
  the_midpoint_of_the_segment(A,B) = A implies A=B
  proof
    assume the_midpoint_of_the_segment(A,B) = A; then
A1: A = 1/2*(A+B) by Th22
     .= 1/2 * A + 1/2 * B by RVSUM_1:51;
A2: 1/2 * A + 1/2 * A = (1/2 + 1/2) * A by RVSUM_1:50
                     .= A by RVSUM_1:52;
    reconsider rA=A, rB=B as Element of REAL 2 by EUCLID:22;
    1/2 * rA + 1/2 * rA = 1/2 * rA + 1/2 * rB by A1,A2;
    then 2*(1/2*A) = 2*(1/2*B) by RVSUM_1:25;
    then (2*1/2)*A = 2*(1/2*B) by RVSUM_1:49;
    then 1*A=1*B by RVSUM_1:49;
    then A = 1*B by RVSUM_1:52;
    hence thesis by RVSUM_1:52;
  end;
