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 Th10:
  o is having_a_unity implies the_unity_wrt (o,D).:A = A -->
  the_unity_wrt o & (o,D).:A is having_a_unity
proof
  given a such that
A1: a is_a_unity_wrt o;
  a = the_unity_wrt o & A --> a is_a_unity_wrt (o,D).:A by A1,Th9,BINOP_1:def 8
;
  hence the_unity_wrt (o,D).:A = A --> the_unity_wrt o by BINOP_1:def 8;
  take A --> a;
  thus thesis by A1,Th9;
end;
