
theorem Th5:
  for A,B being set, a being object holds ~([:A,B:] --> a) = [:B,A:] --> a
proof
  let A,B be set, a be object;
A1: now
    let x be object;
    hereby
      assume x in dom([:B,A:] --> a);
      then consider z,y being object such that
A2:   z in B and
A3:   y in A and
A4:   x = [z,y] by ZFMISC_1:def 2;
      take y,z;
      thus x = [z,y] by A4;
      [y,z] in [:A,B:] by A2,A3,ZFMISC_1:87;
      hence [y,z] in dom([:A,B:] --> a);
    end;
    given y,z being object such that
A5: x = [z,y] and
A6: [y,z] in dom([:A,B:] --> a);
A7: y in A by A6,ZFMISC_1:87;
    z in B by A6,ZFMISC_1:87;
    then x in [:B,A:] by A5,A7,ZFMISC_1:87;
    hence x in dom([:B,A:] --> a);
  end;
  now
    let y,z be object;
    assume
A8: [y,z] in dom([:A,B:] --> a);
    then
A9: y in A by ZFMISC_1:87;
    z in B by A8,ZFMISC_1:87;
    hence ([:B,A:] --> a).(z,y) = a by A9,FUNCOP_1:7,ZFMISC_1:87
      .= ([:A,B:] --> a).(y,z) by A8,FUNCOP_1:7;
  end;
  hence thesis by A1,FUNCT_4:def 2;
end;
