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 Th01:
  for X being non empty set
  for f1,f2,f3 being non empty homogeneous PartFunc of X*, X
   st arity f1 = 0 & arity f2 = 1 & arity f3 = 2
  for S being non empty UAStr
   st the carrier of S = X & <*f1,f2,f3*> c= the charact of S
  holds S is Group-like
  proof
    let X be non empty set;
    let f1,f2,f3 be non empty homogeneous PartFunc of X*, X;
    assume
01: arity f1 = 0;
    assume
02: arity f2 = 1;
    assume
03: arity f3 = 2;
    let S be non empty UAStr;
    assume
04: the carrier of S = X & <*f1,f2,f3*> c= the charact of S;
05: dom <*f1,f2,f3*> = Seg 3 by FINSEQ_2:124;
    hence Seg 3 c= dom the charact of S by 04,RELAT_1:11;
    let f be non empty homogeneous
    PartFunc of (the carrier of S)*, the carrier of S;
    1 in Seg 3 & 2 in Seg 3 & 3 in Seg 3 by FINSEQ_3:1,ENUMSET1:def 1; then
    (the charact of S).1 = <*f1,f2,f3*>.1 &
    (the charact of S).2 = <*f1,f2,f3*>.2 &
    (the charact of S).3 = <*f1,f2,f3*>.3 by 04,05,GRFUNC_1:2;
    hence (f = (the charact of S).1 implies arity f = 0) &
    (f = (the charact of S).2 implies arity f = 1) &
    (f = (the charact of S).3 implies arity f = 2) by 01,02,03;
  end;
