reserve U0 for Universal_Algebra,
  U1 for SubAlgebra of U0,
  o for operation of U0;

theorem
  for H being strict SubAlgebra of U0 for u being Element of U0 holds u
  in (Carr U0).H iff u in H
proof
  let H be strict SubAlgebra of U0;
  let u be Element of U0;
  thus u in (Carr U0).H implies u in H
  proof
A1: H in Sub(U0) by UNIALG_2:def 14;
    assume u in (Carr U0).H;
    then u in the carrier of H by A1,Def4;
    hence thesis by STRUCT_0:def 5;
  end;
  thus u in H implies u in (Carr U0).H
  proof
    H in Sub(U0) by UNIALG_2:def 14;
    then
A2: (Carr U0).H = the carrier of H by Def4;
    assume u in H;
    hence thesis by A2,STRUCT_0:def 5;
  end;
end;
