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 Th15:
  (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) & f
  is_FormalIz_of R & g is_FormalIz_of R implies f*g is_FormalIz_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 holds [[x,y],[f.x,f.y]] in R and
A4: 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
A5: [[g.x,g.y],[(f*g).x,(f*g).y]] in R by A3;
A6: now
    assume g.x = g.y;
    then x = y by FUNCT_2:58;
    hence thesis by A2;
  end;
  [[x,y],[g.x,g.y]] in R by A4;
  hence thesis by A1,A5,A6;
end;
