
theorem Th3:
  for X being set
  for A being Subset of X holds A is pr1(X,X)-binopclosed
proof
  let X be set;
  let A be Subset of X;
  set S = pr1(X,X);
  for x being set holds x in [:A,A:] implies S.x in A
  proof
    let x be set;
    assume x in [:A,A:];
    then consider p,q being object such that
A1: p in A & q in A and
A2: x = [p,q] by ZFMISC_1:def 2;
    S.x = S.(p,q) by A2;
    hence thesis by A1,FUNCT_3:def 4;
  end;
  hence thesis;
end;
