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;
