theorem Th89:
  SignGen(f,B,X) = SignGen(f,B,X/\dom f)
proof
A1: dom SignGen(f,B,X)= dom f =dom SignGen(f,B,X/\dom f) by Def11;
  for i st i in dom SignGen(f,B,X) holds
    SignGen(f,B,X).i = SignGen(f,B,X/\dom f).i
  proof
    let i such that
A2:   i in dom SignGen(f,B,X);
    per cases;
    suppose i in X;
      then SignGen(f,B,X).i = (the_inverseOp_wrt B).(f.i) & i in X/\dom f
        by Def11,A1,XBOOLE_0:def 4,A2;
      hence thesis by Def11,A1,A2;
    end;
    suppose not i in X;
      then SignGen(f,B,X).i = f.i & not i in X/\dom f
        by Def11,XBOOLE_0:def 4,A2;
      hence thesis by Def11,A1,A2;
    end;
  end;
  hence thesis by A1;
end;
