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 Th30:
  (for a,b holds a |^ b = a) implies G is commutative
proof
  assume
A1: for a,b holds a |^ b = a;
  let a,b;
  a |^ b = a by A1;
  hence thesis by Th22;
end;
