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 Th77:
  a in center G iff for b holds a * b = b * a
proof
  thus a in center G implies for b holds a * b = b * a
  proof
    assume a in center G;
    then a in the carrier of center G by STRUCT_0:def 5;
    then a in {b : for c holds b * c = c * b} by Def10;
    then ex b st a = b & for c holds b * c = c * b;
    hence thesis;
  end;
  assume for b holds a * b = b * a;
  then a in {c : for b holds c * b = b * c};
  then a in the carrier of center G by Def10;
  hence thesis by STRUCT_0:def 5;
end;
