reserve x,y for set,
  k,n for Nat,
  i for Integer,
  G for Group,
  a,b,c ,d,e for Element of G,
  A,B,C,D for Subset of G,
  H,H1,H2,H3,H4 for Subgroup of G ,
  N1,N2 for normal Subgroup of G,
  F,F1,F2 for FinSequence of the carrier of G,
  I,I1,I2 for FinSequence of INT;

theorem Th1:
  x in (1).G iff x = 1_G
proof
  thus x in (1).G implies x = 1_G
  proof
    assume x in (1).G;
    then x in the carrier of (1).G by STRUCT_0:def 5;
    then x in {1_G} by GROUP_2:def 7;
    hence thesis by TARSKI:def 1;
  end;
  thus thesis by GROUP_2:46;
end;
