theorem Th11:
  proj1 dom f = proj2 dom ~f & proj2 dom f = proj1 dom ~f
proof
  thus proj1 dom f c= proj2 dom ~f
  proof
    let x be object;
    assume x in proj1 dom f;
    then consider y being object such that
A1: [x,y] in dom f by XTUPLE_0:def 12;
    [y,x] in dom ~f by A1,FUNCT_4:42;
    hence thesis by XTUPLE_0:def 13;
  end;
  thus proj2 dom ~f c= proj1 dom f
  proof
    let y be object;
    assume y in proj2 dom ~f;
    then consider x being object such that
A2: [x,y] in dom ~f by XTUPLE_0:def 13;
    [y,x] in dom f by A2,FUNCT_4:42;
    hence thesis by XTUPLE_0:def 12;
  end;
  thus proj2 dom f c= proj1 dom ~f
  proof
    let y be object;
    assume y in proj2 dom f;
    then consider x being object such that
A3: [x,y] in dom f by XTUPLE_0:def 13;
    [y,x] in dom ~f by A3,FUNCT_4:42;
    hence thesis by XTUPLE_0:def 12;
  end;
  thus proj1 dom ~f c= proj2 dom f
  proof
    let x be object;
    assume x in proj1 dom ~f;
    then consider y being object such that
A4: [x,y] in dom ~f by XTUPLE_0:def 12;
    [y,x] in dom f by A4,FUNCT_4:42;
    hence thesis by XTUPLE_0:def 13;
  end;
end;
