reserve
   a,b,c,x,y,z,A,B,C,X,Y for set,
   f,g for Function,
   V for SetValuation,
   P for Permutation of V,
   p,q,r,s for Element of HP-WFF,
   n for Element of NAT;

theorem Th7:
  for f being involutive Function st rng f c= dom f holds
  f*f = id dom f
  proof
    let f be involutive Function;
    assume rng f c= dom f;
    then
A1: dom(f*f) = dom id dom f by RELAT_1:27;
    now
      let x be object;
      assume
A2:   x in dom(f*f);
      hence (f*f).x = f.(f.x) by FUNCT_1:12
      .= x by A1,A2,PARTIT_2:def 2
      .= (id dom f).x by A1,A2,FUNCT_1:18;
    end;
    hence thesis by A1, FUNCT_1:2;
  end;
