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 Th27:
  r <= 1 implies dist(p,r*p+(1-r)*q) = (1-r)*dist(p,q)
proof
  assume r <= 1;
  then 1+ r <= 1 + 1 by XREAL_1:6;
  then 1-r >= 1-1 by XREAL_1:21;
  then
A1: |.1-r.| = 1-r by ABSVALUE:def 1;
  thus dist(p,r*p+(1-r)*q) = dist((r+(1-r))*p,r*p+(1-r)*q) by RLVECT_1:def 8
    .= dist(r*p+(1-r)*p,r*p+(1-r)*q) by RLVECT_1:def 6
    .= dist((1-r)*p,(1-r)*q) by Th21
    .= (1-r)*dist(p,q) by A1,Th26;
end;
