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 Th18:
  f is_automorphism_of R & g is_automorphism_of R implies g*f
  is_automorphism_of R
proof
  assume that
A1: for x,y,z,t holds ([[x,y],[z,t]] in R iff [[f.x,f.y],[f.z,f.t]] in R ) and
A2: for x,y,z,t holds ([[x,y],[z,t]] in R iff [[g.x,g.y],[g.z,g.t]] in R );
  let x,y,z,t;
A3: g.(f.x) = (g*f).x & g.(f.y) = (g*f).y by FUNCT_2:15;
A4: g.(f.z) = (g*f).z & g.(f.t) = (g*f).t by FUNCT_2:15;
  [[x,y],[z,t]] in R iff [[f.x,f.y],[f.z,f.t]] in R by A1;
  hence thesis by A2,A3,A4;
end;
