reserve x,y,y1,y2 for set;
reserve G for Group;
reserve a,b,c,d,g,h for Element of G;
reserve A,B,C,D for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve n for Nat;
reserve i for Integer;

theorem Th22:
  a |^ b = a iff a * b = b * a
proof
  thus a |^ b = a implies a * b = b * a
  proof
    assume a |^ b = a;
    then a = b" * (a * b) by GROUP_1:def 3;
    hence thesis by GROUP_1:13;
  end;
  assume a * b = b * a;
  then a = b" * (a * b) by GROUP_1:13;
  hence thesis by GROUP_1:def 3;
end;
