reserve x,y for set,
  k,n for Nat,
  i for Integer,
  G for Group,
  a,b,c ,d,e for Element of G,
  A,B,C,D for Subset of G,
  H,H1,H2,H3,H4 for Subgroup of G ,
  N1,N2 for normal Subgroup of G,
  F,F1,F2 for FinSequence of the carrier of G,
  I,I1,I2 for FinSequence of INT;

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;
