reserve x,y,z, X,Y,Z for set,
  n for Element of NAT;
reserve A for set,
  D for non empty set,
  a,b,c,l,r for Element of D,
  o,o9 for BinOp of D,
  f,g,h for Function of A,D;

theorem Th9:
  a is_a_unity_wrt o implies A --> a is_a_unity_wrt (o,D).:A
proof
  set e = A --> a;
  set F = (o,D).:A;
  assume
A1: a is_a_unity_wrt o;
  now
    let f be Element of Funcs(A,D);
A2: dom f = A by FUNCT_2:def 1;
    thus F.(e,f) = o.:(e,f) by Def2
      .= o[;](a,f) by A2,FUNCOP_1:31
      .= f by A1,Th5;
    thus F.(f,e) = o.:(f,e) by Def2
      .= o[:](f,a) by A2,FUNCOP_1:26
      .= f by A1,Th5;
  end;
  hence thesis by BINOP_1:3;
end;
