reserve X for MetrSpace,
  x,y,z for Element of X,
  A for non empty set,
  G for Function of [:A,A:],REAL,
  f for Function,
  k,n,m,m1,m2 for Nat,
  q,r for Real;

theorem Th1:
  |.dist(x,z) - dist(y,z).| <= dist(x,y)
proof
  dist(y,z) <= dist(y,x) + dist(x,z) by METRIC_1:4;
  then dist(y,z) - dist(x,z) <= dist(x,y) by XREAL_1:20;
  then
A1: - dist(x,y) <= - (dist(y,z) - dist(x,z)) by XREAL_1:24;
  dist(x,z) <= dist(x,y) + dist(y,z) by METRIC_1:4;
  then dist(x,z) - dist(y,z) <= dist(x,y) by XREAL_1:20;
  hence thesis by A1,ABSVALUE:5;
end;
