reserve X for set,
  Y for non empty set;
reserve n for Nat;
reserve r for Real,
  M for non empty MetrSpace;
reserve n for Nat,
  p,q,q1,q2 for Point of TOP-REAL 2,
  r,s1,s2,t1,t2 for Real,
  x,y for Point of Euclid 2;

theorem Th28:
  0 <= r implies dist(q,r*p+(1-r)*q) = r*dist(p,q)
proof
  assume 0 <= r;
  then
A1: |.r.| = r by ABSVALUE:def 1;
  thus dist(q,r*p+(1-r)*q) = dist(r*p+(1-r)*q,(r+(1-r))*q) by RLVECT_1:def 8
    .= dist(r*q+(1-r)*q,r*p+(1-r)*q) by RLVECT_1:def 6
    .= dist(r*q,r*p) by Th21
    .= r*dist(p,q) by A1,Th26;
end;
