theorem Th7:
  for G being strict Group, B being strict Subgroup of G holds
  G` is Subgroup of B iff
    for a,b being Element of G holds [.a,b.] in B
proof
  defpred P[set,set] means not contradiction;
  let G be strict Group, B be strict Subgroup of G;
  thus G` is Subgroup of B implies for a,b being Element of G holds
  [.a,b.] in B by GROUP_2:40,GROUP_5:74;
  deffunc F(Element of G,Element of G) = [.$1,$2.];
  reconsider X = {F(a,b) where a is Element of G, b is Element of G : P[a,b]}
  as Subset of G from DOMAIN_1:sch 9;
  assume
A1: for a,b being Element of G holds [.a,b.] in B;
  X c= the carrier of B
  proof
    let x be object;
    assume x in X;
    then ex a,b being Element of G st x = [.a,b.];
    then x in B by A1;
    hence thesis;
  end;
  then gr X is Subgroup of B by GROUP_4:def 4;
  hence thesis by Th6;
end;
