
theorem
  for A,B being non empty set
  for f being Covariant Contravariant bifunction of A,B
  ex b being Element of B st f = [:A,A:] --> [b,b]
proof
  let A,B be non empty set;
  let f be Covariant Contravariant bifunction of A,B;
  consider g1 being Function of A,B such that
A1: f = [:g1,g1:] by Def1;
  consider g2 being Function of A,B such that
A2: f = ~[:g2,g2:] by Def2;
  set a = the Element of A;
  take b = g1.a;
A3: dom f = [:A,A:] by FUNCT_2:def 1;
  now
    let z be object;
    assume z in dom f;
    then consider a1,a2 being Element of A such that
A4: z = [a1,a2] by DOMAIN_1:1;
A5: dom g2 = A by FUNCT_2:def 1;
A6: dom g1 = A by FUNCT_2:def 1;
A7: dom[:g2,g2:] = [:dom g2, dom g2:] by FUNCT_3:def 8;
    then
A8: [a1,a] in dom[:g2,g2:] by A5,ZFMISC_1:87;
A9: dom g2 = A by FUNCT_2:def 1;
    [b,g1.a1] = f.(a,a1) by A1,A6,FUNCT_3:def 8
      .= [:g2,g2:].(a1,a) by A2,A8,FUNCT_4:def 2
      .= [g2.a1,g2.a] by A9,FUNCT_3:def 8;
    then
A10: g2.a1 = b by XTUPLE_0:1;
A11: [a2,a] in dom[:g2,g2:] by A5,A7,ZFMISC_1:87;
    [b,g1.a2] = f.(a,a2) by A1,A6,FUNCT_3:def 8
      .= [:g2,g2:].(a2,a) by A2,A11,FUNCT_4:def 2
      .= [g2.a2,g2.a] by A9,FUNCT_3:def 8;
    then
A12: g2.a2 = b by XTUPLE_0:1;
A13: [a2,a1] in dom[:g2,g2:] by A5,A7,ZFMISC_1:87;
    thus f.z = [:g1,g1:].(a1,a2) by A1,A4
      .= [:g2,g2:].(a2,a1) by A1,A2,A13,FUNCT_4:def 2
      .= [b,b] by A9,A10,A12,FUNCT_3:def 8;
  end;
  hence thesis by A3,FUNCOP_1:11;
end;
