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_transitive_in [:A,A:] & f is_FormalIz_of R & g is_FormalIz_of R
  implies (f*g) is_FormalIz_of R
proof
  assume that
A1: 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
A2: for x,y holds [[x,y],[f.x,f.y]] in R and
A3: for x,y holds [[x,y],[g.x,g.y]] in R;
  let x,y;
  f.(g.x) = (f*g).x & f.(g.y) = (f*g).y by FUNCT_2:15;
  then
A4: [[g.x,g.y],[(f*g).x,(f*g).y]] in R by A2;
A5: [(f*g).x,(f*g).y] in [:A,A:] by ZFMISC_1:def 2;
A6: [x,y] in [:A,A:] & [g.x,g.y] in [:A,A:] by ZFMISC_1:def 2;
  [[x,y],[g.x,g.y]] in R by A3;
  hence thesis by A1,A4,A6,A5;
end;
