theorem Th113:
  s1 is_equivalent_with s1
proof
  per cases;
  suppose
    s1 is empty;
    hence thesis;
  end;
  suppose
A1: s1 is not empty;
    set f1=the_series_of_quotients_of s1;
    now
      set p = id dom f1;
      reconsider p as Function of dom f1,dom f1;
      p is onto;
      then reconsider p as Permutation of dom f1;
      take p;
A2:   now
        let H1,H2 be GroupWithOperators of O;
        let i,j;
        assume
A3:     i in dom f1 & j=p".i;
A4:     p" = p by FUNCT_1:45;
        assume H1=f1.i & H2=f1.j;
        hence H1,H2 are_isomorphic by A3,A4,FUNCT_1:18;
      end;
      thus f1,f1 are_equivalent_under p,O by A2;
    end;
    hence thesis by A1,Th108;
  end;
end;
