reserve X for non empty UNITSTR;
reserve a, b for Real;
reserve x, y for Point of X;
reserve X for RealUnitarySpace;
reserve x, y, z, u, v for Point of X;

theorem
  x <> y iff dist(x,y) > 0
proof
  thus x <> y implies dist(x,y) > 0
  proof
    assume x <> y;
    then dist(x,y) <> 0 by Th36;
    hence thesis by Th28;
  end;
  thus thesis by Th34;
end;
