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 Th36:
  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,Th26;
    hence contradiction by A1,RLVECT_1:21;
  end;
  thus thesis by Th34;
end;
