 reserve f,g for Function;
 reserve R for non empty reflexive RelStr;
 reserve R for non empty RelStr;

theorem FlipCompose:
  for f,g being map of R holds
    Flip (f * g) = Flip f * Flip g
  proof
    let f,g be map of R;
    set fg = Flip (f * g);
    set ff = Flip f;
    set gg = Flip g;
    for x being Subset of R holds fg.x = (ff * gg).x
    proof
      let x be Subset of R;
      x` in bool the carrier of R; then
A1:   x` in dom g by FUNCT_2:def 1;
a2:   dom Flip g = bool the carrier of R by FUNCT_2:def 1;
      fg.x = ((f * g).x`)` by ROUGHS_2:def 14
        .= (f.((g.x`)`)`)` by FUNCT_1:13,A1
        .= ff.((g.x`)`) by ROUGHS_2:def 14
        .= ff.(gg.x) by ROUGHS_2:def 14
        .= (ff * gg).x by a2,FUNCT_1:13;
      hence thesis;
    end;
    hence thesis;
  end;
