reserve U1,U2,U3 for Universal_Algebra,
  n,m for Nat,
  o1 for operation of U1,
  o2 for operation of U2,
  o3 for operation of U3,
  x,y for set;

theorem Th1:
  for B be non empty Subset of U1 st B = the carrier of U1 holds
  Opers(U1,B) = the charact of(U1)
proof
  let B be non empty Subset of U1;
A1: dom Opers(U1,B) = dom the charact of(U1) by UNIALG_2:def 6;
  assume
A2: B = the carrier of U1;
  now
    let n be Nat;
    assume
A3: n in dom the charact of(U1);
    then reconsider o = (the charact of U1).n as operation of U1 by
FUNCT_1:def 3;
    thus Opers(U1,B).n = o/.B by A1,A3,UNIALG_2:def 6
      .= (the charact of U1).n by A2,UNIALG_2:4;
  end;
  hence thesis by A1;
end;
