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 (R5) (R8) implies (1.S)" <<>> 1.S
  proof
    assume
A1: S is (R5) (R8);
    take (1.S)"*1.S;
    thus (1.S)"*1.S =*=> (1.S)" by A1,Th2;
    thus (1.S)" * 1.S =*=> 1.S by A1,Th2;
  end;
