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 Th32:
  A |^ B <> {} iff A <> {} & B <> {}
proof
  set x = the Element of A;
  set y = the Element of B;
  thus A |^ B <> {} implies A <> {} & B <> {}
  proof
    set x = the Element of A |^ B;
    assume A |^ B <> {};
    then ex a,b st x = a |^ b & a in A & b in B by Th31;
    hence thesis;
  end;
  assume
A1: A <> {};
  assume
A2: B <> {};
  then reconsider x,y as Element of G by A1,TARSKI:def 3;
  x |^ y in A |^ B by A1,A2;
  hence thesis;
end;
