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 ThI3:
  a ==> b implies c*a ==> c*b
  proof
    assume
A0: a ==> b;
    set o = In(3, dom the charact of S);
    arity Den(o, S) = 2 by ThB; then
    dom Den(o, S) = 2-tuples_on the carrier of S by MARGREL1:22; then
    reconsider ac = <*c,a*>, bc = <*c,b*> as Element of dom Den(o, S)
    by FINSEQ_2:101;
A2: dom <*c,a*> = Seg 2 & 2 in Seg 2 by FINSEQ_1:1,89;
A3: <*c,a*>.2 = a;
    <*c,a*>+*(2,b) = <*c,b*> by COMPUT_1:1; then
    Den(o,S).ac ==> Den(o,S).bc by A0,A2,A3,DEF2;
    hence c*a ==> c*b;
  end;
