 reserve X,a,b,c,x,y,z,t for set;
 reserve R for Relation;

theorem FlipCC:
  for A being non empty set
  for f, g being Function of bool A, bool A st
    f cc= g holds
      Flip g cc= Flip f
  proof
    let A be non empty set;
    let f, g be Function of bool A, bool A;
    assume
A1: f cc= g;
    set ff = Flip f, gg = Flip g;
a2: dom ff = bool A by FUNCT_2:def 1;
    for x being set st x in dom gg holds gg.x c= ff.x
    proof
      let x be set;
      assume x in dom gg; then
      reconsider xx = x as Subset of A;
B1:   (Flip g).xx = (g.xx`)` by ROUGHS_2:def 14;
B2:   (Flip f).xx = (f.xx`)` by ROUGHS_2:def 14;
      dom f = bool A by FUNCT_2:def 1; then
      f.xx` c= g.xx` by A1,ALTCAT_2:def 1;
      hence thesis by B1,B2,SUBSET_1:12;
    end;
    hence thesis by a2,ALTCAT_2:def 1;
  end;
