theorem Th47:
  x <> y iff dist(x,y) <> 0
proof
  thus x <> y implies dist(x,y) <> 0
  proof
    assume that
A1: x <> y and
A2: dist(x,y) = 0;
    x - y = 09(X) by A2,Th37;
    hence contradiction by A1,RLVECT_1:21;
  end;
  thus thesis by Th45;
end;
