theorem Th52:
  x in commutators(H1,H2) iff ex a,b st x = [.a,b.] & a in H1 & b in H2
proof
  thus x in commutators(H1,H2) implies ex a,b st x = [.a,b.] & a in H1 & b in
  H2
  proof
    assume x in commutators(H1,H2);
    then consider a,b such that
A1: x = [.a,b.] and
A2: a in carr H1 & b in carr H2;
    a in H1 & b in H2 by A2,STRUCT_0:def 5;
    hence thesis by A1;
  end;
  given a,b such that
A3: x = [.a,b.] and
A4: a in H1 & b in H2;
  a in carr H1 & b in carr H2 by A4,STRUCT_0:def 5;
  hence thesis by A3;
end;
