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;
reserve N1,N2 for Subgroup of G;

theorem
  N1 is Subgroup of N2 implies N1 ~ N1 c= N2 ~ N2
proof
  assume
A1: N1 is Subgroup of N2; then
A2: N2 ~ N1 c= N2 ~ N2 by Th56;
    N1 ~ N1 c= N2 ~ N1 by A1,Th57;
  hence thesis by A2;
end;
