reserve A for non empty set,
  a,b,x,y,z,t for Element of A,
  f,g,h for Permutation of A;
reserve R for Relation of [:A,A:];

theorem
  (for a,b,x,y,z,t st [[x,y],[a,b]] in R & [[a,b],[z,t]] in R & a<>b
  holds [[x,y],[z,t]] in R) & (for x,y,z holds [[x,x],[y,z]] in R) & R
  is_symmetric_in [:A,A:] & f is_FormalIz_of R implies f is_automorphism_of R
proof
  assume that
A1: for a,b,x,y,z,t st [[x,y],[a,b]] in R & [[a,b],[z,t]] in R & a<>b
  holds [[x,y],[z,t]] in R and
A2: for x,y,z holds [[x,x],[y,z]] in R and
A3: for x,y being object st x in [:A,A:] & y in [:A,A:] & [x,y] in R holds
  [y,x] in R and
A4: for x,y holds [[x,y],[f.x,f.y]] in R;
A5: for x,y,z holds [[y,z],[x,x]] in R
  proof
    let x,y,z;
A6: [y,z] in [:A,A:] & [x,x] in [:A,A:] by ZFMISC_1:def 2;
    [[x,x],[y,z]] in R by A2;
    hence thesis by A3,A6;
  end;
  let x,y,z,t;
A7: [[x,y],[f.x,f.y]] in R by A4;
A8: [[z,t],[f.z,f.t]] in R by A4;
  [z,t] in [:A,A:] & [f.z,f.t] in [:A,A:] by ZFMISC_1:def 2;
  then
A9: [[f.z,f.t],[z,t]] in R by A3,A8;
A10: now
    assume
A11: [[f.x,f.y],[f.z,f.t]] in R;
A12: now
      assume that
A13:  f.x<>f.y and
A14:  f.z<>f.t;
      [[x,y],[f.z,f.t]] in R by A1,A7,A11,A13;
      hence [[x,y],[z,t]] in R by A1,A9,A14;
    end;
A15: now
      assume f.z = f.t;
      then z=t by FUNCT_2:58;
      hence [[x,y],[z,t]] in R by A5;
    end;
    now
      assume f.x = f.y;
      then x=y by FUNCT_2:58;
      hence [[x,y],[z,t]] in R by A2;
    end;
    hence [[x,y],[z,t]] in R by A15,A12;
  end;
  [x,y] in [:A,A:] & [f.x,f.y] in [:A,A:] by ZFMISC_1:def 2;
  then
A16: [[f.x,f.y],[x,y]] in R by A3,A7;
  now
    assume
A17: [[x,y],[z,t]] in R;
    now
      assume that
A18:  x<>y and
A19:  z<>t;
      [[f.x,f.y],[z,t]] in R by A1,A16,A17,A18;
      hence [[f.x,f.y],[f.z,f.t]] in R by A1,A8,A19;
    end;
    hence [[f.x,f.y],[f.z,f.t]] in R by A2,A5;
  end;
  hence thesis by A10;
end;
