theorem Lem11A:
  for t for xi being Element of dom t st t.xi = [x,s]
  holds dom t = dom (t with-replacement(xi,x1-term))
  proof
    let t; set t1 = x1-term;
    let xi be Element of dom t;
    assume t.xi = [x,s];
    then
A0: dom t c= dom (t with-replacement(xi,x1-term)) by Lem11;
    dom (t with-replacement(xi,x1-term)) c= dom t
    proof let a;
      assume a in dom (t with-replacement(xi,x1-term));
      then reconsider q = a as Element of dom (t with-replacement(xi,x1-term));
      dom (t with-replacement(xi,x1-term)) = dom t with-replacement(xi, dom t1)
      by TREES_2:def 11;
      then per cases by TREES_1:def 9;
      suppose q in dom t;
        hence a in dom t;
      end;
      suppose ex r being FinSequence of NAT st r in dom t1 & q = xi^r;
        then consider r being FinSequence of NAT such that
A1:     r in dom t1 & q = xi^r;
        r in {{}} by A1,TREES_1:29;
        then r = {};
        hence a in dom t by A1;
      end;
    end;
    hence thesis by A0,XBOOLE_0:def 10;
  end;
