reserve A for non empty set,
  a,b,x,y,z,t for Element of A,
  f,g,h for Permutation of A;

theorem Th2:
  f.x=y iff (f").y=x
proof
A1: now
    x in A;
    then
A2: x in dom f by FUNCT_2:def 1;
    assume f.x=y;
    hence x=f".y by A2,FUNCT_1:34;
  end;
  rng f = A by FUNCT_2:def 3;
  hence thesis by A1,FUNCT_1:35;
end;
