theorem
  dom f c= X implies SignGen(f,B,X) = (the_inverseOp_wrt B)*f
proof
  assume
A1: dom f c= X;
  rng f c= D=dom (the_inverseOp_wrt B) by FUNCT_2:def 1;
  then
A2: dom ((the_inverseOp_wrt B)*f) = dom f = dom SignGen(f,B,X)
    by Def11,RELAT_1:27;
  for k st k in dom SignGen(f,B,X) holds SignGen(f,B,X).k=
    ((the_inverseOp_wrt B)*f).k
  proof
    let k such that
A3:   k in dom SignGen(f,B,X);
    SignGen(f,B,X).k = (the_inverseOp_wrt B).(f.k) by A1,A3,A2,Def11;
    hence thesis by A3,A2,FUNCT_1:12;
  end;
  hence thesis by A2;
end;
