reserve G for Group;
reserve A,B for non empty Subset of G;
reserve N,H,H1,H2 for Subgroup of G;
reserve x,a,b for Element of G;

theorem Th6:
  x in H1 * H2 iff ex a,b st x = a * b & a in H1 & b in H2
proof
  thus x in H1 * H2 implies ex a,b st x = a * b & a in H1 & b in H2
  proof
    assume x in H1 * H2;
    then consider a,b such that
A1: x = a * b & a in carr(H1) & b in carr(H2);
    a in H1 & b in H2 by A1,STRUCT_0:def 5;
    hence thesis by A1;
  end;
  given a,b such that
A2: x = a * b & a in H1 & b in H2;
  a in carr(H1) & b in carr(H2) by A2,STRUCT_0:def 5;
  hence thesis by A2;
end;
