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) (R2) (R3) implies a" * (a * b) <<>> b
  proof
    assume
A1: S is (R1) (R2) (R3);
    take (a"*a)*b;
    thus (a"*a)*b =*=> a"*(a*b) by A1,Th2;
    (a"*a)*b ==> 1.S * b & 1.S * b ==> b by A1,ThI2; then
    (a"*a)*b =*=> 1.S * b & 1.S * b =*=> b by Th2;
    hence (a"*a)*b =*=> b by Th3;
  end;
