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 Th13:
  R is_symmetric_in [:A,A:] & f is_FormalIz_of R implies f" is_FormalIz_of R
proof
  assume
A1: for x,y being object st x in [:A,A:] & y in [:A,A:] & [x,y] in R holds
  [y,x] in R;
  assume
A2: for x,y holds [[x,y],[f.x,f.y]] in R;
  let x,y;
A3: [[f".x,f".y],[f.(f".x),f.(f".y)]] in R by A2;
A4: [f".x,f".y] in [:A,A:] & [f.(f".x),f.(f".y)] in [:A,A:] by ZFMISC_1:def 2;
  f.(f".x) = x & f.(f".y) =y by Th2;
  hence thesis by A1,A3,A4;
end;
