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;

theorem
  a ==> b implies a" ==> b"
  proof
    assume
A0: a ==> b;
    set o = In(2, dom the charact of S);
    arity Den(o, S) = 1 by ThB; then
    dom Den(o, S) = 1-tuples_on the carrier of S by MARGREL1:22; then
    reconsider aa = <*a*>, bb = <*b*> as Element of dom Den(o, S)
    by FINSEQ_2:98;
A2: dom <*a*> = Seg 1 & 1 in Seg 1 by FINSEQ_1:1,38;
A3: <*a*>.1 = a;
    <*a*>+*(1,b) = <*b*> by FUNCT_7:95; then
    Den(o,S).aa ==> Den(o,S).bb by A0,A2,A3,DEF2;
    hence a" ==> b";
  end;
