reserve U for Universe;
reserve x for Element of U;
reserve U1,U2 for Universe;

theorem
  (union the set of all the carrier of Trivial-addLoopStr(x) where
  x is Element of U) = U
  proof
    set T = the set of all the carrier of Trivial-addLoopStr(x) where
    x is Element of U;
    now
      hereby
        let o be object;
        assume o in union T;
        then consider y be set such that
A1:     o in y and
A2:     y in T by TARSKI:def 4;
        consider x be Element of U such that
A3:     y = the carrier of Trivial-addLoopStr(x) by A2;
        o = x by A3,A1,TARSKI:def 1;
        hence o in U;
      end;
      hence union T c= U;
      hereby
        let o be object;
        assume o in U;
        then reconsider x = o as Element of U;
        {x} is Element of U;
        then reconsider y = the carrier of Trivial-addLoopStr(x)
          as Element of U;
        o in y in T;
        hence o in union T by TARSKI:def 4;
      end;
      hence U c= union T;
    end;
    hence thesis;
  end;
