reserve U0,U1,U2,U3 for Universal_Algebra,
  n for Nat,
  x,y for set;
reserve A for non empty Subset of U0,
  o for operation of U0,
  x1,y1 for FinSequence of A;

theorem Th14:
  for A be non empty Subset of U0 holds UAStr (#A,Opers(U0,A)#) is
  strict Universal_Algebra
proof
  let A be non empty Subset of U0;
  set C = UAStr(#A,Opers(U0,A)#);
A1: dom the charact of(C) = dom the charact of(U0) by Def6;
  for n be object st n in dom the charact of(C) holds (the charact of C).n
  is non empty
  proof
    let n be object;
    assume
A2: n in dom the charact of(C);
    then reconsider o = (the charact of U0).n as operation of U0 by A1,
FUNCT_1:def 3;
    (the charact of C).n =o/.A by A2,Def6;
    hence thesis;
  end;
  then
A3: the charact of(C) is non-empty by FUNCT_1:def 9;
  for n be Nat ,h be PartFunc of C st n in dom the charact of(C) & h = (
  the charact of C).n holds h is quasi_total
  proof
    let n;
    let h be PartFunc of (the carrier of C)*,the carrier of C;
    assume that
A4: n in dom the charact of(C) and
A5: h = (the charact of C).n;
    reconsider o = (the charact of U0).n as operation of U0 by A1,A4,
FUNCT_1:def 3;
    h =o/.A by A4,A5,Def6;
    hence thesis;
  end;
  then
A6: the charact of C is quasi_total;
  for n be Nat ,h be PartFunc of (the carrier of C)*,the carrier of C st n
  in dom the charact of(C) & h = (the charact of C).n holds h is homogeneous
  proof
    let n;
    let h be PartFunc of (the carrier of C)*,the carrier of C;
    assume that
A7: n in dom the charact of(C) and
A8: h = (the charact of C).n;
    reconsider o = (the charact of U0).n as operation of U0 by A1,A7,
FUNCT_1:def 3;
    h =o/.A by A7,A8,Def6;
    hence thesis;
  end;
  then
A9: the charact of (C) is homogeneous;
  the charact of C <> {} by A1,RELAT_1:38,41;
  hence thesis by A9,A6,A3,UNIALG_1:def 1,def 2,def 3;
end;
