reserve n for Element of NAT;
reserve i for Integer;
reserve G,H,I for Group;
reserve A,B for Subgroup of G;
reserve N for normal Subgroup of G;
reserve a,a1,a2,a3,b,b1 for Element of G;
reserve c,d for Element of H;
reserve f for Function of the carrier of G, the carrier of H;
reserve x,y,y1,y2,z for set;
reserve A1,A2 for Subset of G;

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;
