reserve X for set,
        D for a_partition of X,
        TG for non empty TopologicalGroup;
reserve A for Subset of X;
reserve US for UniformSpace;
reserve R for Relation of X;

theorem Th27:
  for X being set, R being total transitive Relation of X holds
  uniformity_induced_by(R) is axiom_U3
  proof
    let X be set, R be total transitive Relation of X;
A1: rho(R) is axiom_UP3 by Th24;
    let S be Element of the entourages of uniformity_induced_by(R);
    reconsider S1 = S as Element of rho(R);
    consider W be Element of rho(R) such that
A2: W * W c= S1 by A1;
    thus ex W be Element of the entourages of uniformity_induced_by(R) st
      W * W c= S by A2;
  end;
