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
  R is_symmetric_in [:A,A:] & R is_transitive_in [:A,A:] & f
  is_FormalIz_of R implies f is_automorphism_of R
proof
  assume that
A1: for x,y being object st x in [:A,A:] & y in [:A,A:] & [x,y] in R holds
  [y,x] in R and
A2: for x,y,z being object st x in [:A,A:] & y in [:A,A:] & z in [:A,A:] &
  [x,y] in R & [y,z] in R holds [x,z] in R and
A3: for x,y holds [[x,y],[f.x,f.y]] in R;
  let x,y,z,t;
A4: [z,t] in [:A,A:] by ZFMISC_1:def 2;
A5: [f.z,f.t] in [:A,A:] by ZFMISC_1:def 2;
A6: [f.x,f.y] in [:A,A:] by ZFMISC_1:def 2;
A7: [x,y] in [:A,A:] by ZFMISC_1:def 2;
A8: now
    [[z,t],[f.z,f.t]] in R by A3;
    then
A9: [[f.z,f.t],[z,t]] in R by A1,A4,A5;
    assume
A10: [[f.x,f.y],[f.z,f.t]] in R;
    [[x,y],[f.x,f.y]] in R by A3;
    then [[x,y],[f.z,f.t]] in R by A2,A7,A6,A5,A10;
    hence [[x,y],[z,t]] in R by A2,A7,A4,A5,A9;
  end;
  now
    [[x,y],[f.x,f.y]] in R by A3;
    then
A11: [[f.x,f.y],[x,y]] in R by A1,A7,A6;
A12: [[z,t],[f.z,f.t]] in R by A3;
    assume [[x,y],[z,t]] in R;
    then [[f.x,f.y],[z,t]] in R by A2,A7,A4,A6,A11;
    hence [[f.x,f.y],[f.z,f.t]] in R by A2,A4,A6,A5,A12;
  end;
  hence thesis by A8;
end;
