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

theorem Th80:
  ex y being Element of U st GO y, Trivial-addLoopStr(x)
  proof
    set Tx = Trivial-addLoopStr(x);
    reconsider x1 = {x} as Element of U;
    reconsider x2 = op2(x) as Element of U by Th78;
    reconsider x3 = comp Tx as Element of U by Th79;
    reconsider x4 = op0(x) as Element of U;
    reconsider y = [x1,x2,x3,x4] as Element of U;
    now
      take y;
      thus y = [x1,x2,x3,x4];
      x4 = 0.Tx by TARSKI:def 1;
      hence ex G being strict AddGroup st Trivial-addLoopStr(x) = G &
      x1 = the carrier of G & x2 = the addF of G & x3 = comp G & x4 = 0.G;
    end;
    hence thesis by GRCAT_1:def 23;
  end;
