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
  U0 is SubAlgebra of U0
proof
A1: for B be non empty Subset of U0 st B=the carrier of U0 holds the charact
  of(U0) = Opers(U0,B) & B is opers_closed
  proof
    let B be non empty Subset of U0;
    assume
A2: B =the carrier of U0;
A3: dom the charact of(U0) = dom Opers(U0,B) by Def6;
    for n be Nat st n in dom the charact of(U0) holds (the charact of(U0))
    .n = (Opers(U0,B)).n
    proof
      let n be Nat;
      assume
A4:   n in dom the charact of(U0);
      then reconsider o =(the charact of(U0)).n as operation of U0 by
FUNCT_1:def 3;
      (Opers(U0,B)).n = o/.B by A3,A4,Def6;
      hence thesis by A2,Th4;
    end;
    hence thesis by A2,A3,Th4,FINSEQ_1:13;
  end;
  the carrier of U0 c= the carrier of U0;
  hence thesis by A1,Def7;
end;
