reserve X for ARS, a,b,c,u,v,w,x,y,z for Element of X;
reserve i,j,k for Element of ARS_01;
reserve l,m,n for Element of ARS_02;
reserve A for set;
reserve
  S for Group-like quasi_total partial invariant non empty non-empty TRSStr;
reserve a,b,c for Element of S;
reserve S for Group-like quasi_total partial non empty non-empty TRSStr;
reserve a,b,c for Element of S;
reserve
  S for Group-like quasi_total partial invariant non empty non-empty TRSStr,
  a,b,c for Element of S;

theorem
  S is (R1) (R4) implies (1.S)" * a <<>> a
  proof
    assume
A1: S is (R1) (R4);
    take (1.S)"*(1.S*a);
    1.S*a ==> a by A1;
    hence (1.S)"*(1.S*a) =*=> (1.S)" * a by Th2,ThI3;
    thus thesis by A1,Th2;
  end;
