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;

theorem ThB:
  for S being partial non empty non-empty UAStr st S is Group-like
  holds
    arity Den(In(1, dom the charact of S), S) = 0 &
    arity Den(In(2, dom the charact of S), S) = 1 &
    arity Den(In(3, dom the charact of S), S) = 2
  proof
    let S be partial non empty non-empty UAStr;
    assume
A1: S is Group-like; then
    1 is OperSymbol of S & 2 is OperSymbol of S & 3 is OperSymbol of S
    by ThA; then
    In(1, dom the charact of S) = 1 &
    In(2, dom the charact of S) = 2 &
    In(3, dom the charact of S) = 3;
    hence thesis by A1,PUA2MSS1:def 1;
  end;
