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 Th02:
  for X being non empty set
  for f1,f2,f3 being non empty quasi_total homogeneous PartFunc of X*, X
  for S being non empty UAStr
   st the carrier of S = X & <*f1,f2,f3*> = the charact of S
  holds S is quasi_total partial
  proof
    let X be non empty set;
    let f1,f2,f3 be non empty quasi_total homogeneous PartFunc of X*, X;
    let S be non empty UAStr;
    assume
04: the carrier of S = X & <*f1,f2,f3*> = the charact of S;
    set A = the carrier of S;
    thus S is quasi_total
    proof
      let i be Nat, h being PartFunc of A*,A;
      assume i in dom the charact of S; then
      i in Seg 3 by 04,FINSEQ_1:89; then
      i = 1 or i = 2 or i = 3 by FINSEQ_3:1,ENUMSET1:def 1;
      hence thesis by 04;
    end;
    let i be Nat, h being PartFunc of A*,A;
    assume i in dom the charact of S; then
    i in Seg 3 by 04,FINSEQ_1:89; then
    i = 1 or i = 2 or i = 3 by FINSEQ_3:1,ENUMSET1:def 1;
    hence thesis by 04;
  end;
