theorem Lem10:
  for t,t1 for xi being Element of dom t st
  t1 = t with-replacement(xi,x-term) & t is x-omitting holds t1 is context of x
  proof
    let t,t1;
    let xi be Element of dom t;
    assume Z1: t1 = t with-replacement(xi,x-term);
    assume Z2: t is x-omitting;
    Coim(t1,[x,s]) = {xi}
    proof
      thus Coim(t1,[x,s]) c= {xi}
      proof
        let a; assume
A0:     a in Coim(t1,[x,s]);
        then
A1:     a in dom t1 & t1.a in {[x,s]} by FUNCT_1:def 7;
        reconsider nu = a as Element of dom t1 by A0,FUNCT_1:def 7;
        nu in dom t1;
        then
A5:     xi in dom t & nu in dom t with-replacement(xi, dom (x-term))
        by Z1,TREES_2:def 11;
        then per cases by Z1,TREES_2:def 11;
        suppose
A3:       t1.nu = t.nu & not xi is_a_prefix_of nu;
          then not ex r being FinSequence of NAT st r in dom (x-term) &
          nu = xi^r by TREES_1:1;
          then [x,s] in {[x,s]} & nu in dom t by A5,TARSKI:def 1,TREES_1:def 9;
          hence thesis by Z2,A3,A1,FUNCT_1:def 7;
        end;
        suppose ex r being FinSequence of NAT st r in dom (x-term) & nu = xi^r
          & t1.nu = (x-term).r;
          then consider r being FinSequence of NAT such that
A6:       r in dom (x-term) & nu = xi^r & t1.nu = (x-term).r;
          r in {{}} by A6,TREES_1:29;
          then r = {};
          hence thesis by A6,TARSKI:def 1;
        end;
      end;
      let a; assume a in {xi};
      then
A7:   a = xi by TARSKI:def 1;
A9:   xi in dom t with-replacement(xi, dom(x-term)) = dom t1
      by Z1,TREES_1:def 9,TREES_2:def 11;
      then consider r being FinSequence of NAT such that
A8:   r in dom (x-term) & xi = xi^r & t1.xi = (x-term).r by Z1,TREES_2:def 11;
      r = {} by A8,FINSEQ_1:87;
      then t1.xi = [x,s] in {[x,s]} by A8,TARSKI:def 1,TREES_4:3;
      hence thesis by A7,A9,FUNCT_1:def 7;
    end;
    then card Coim(t1,[x,s]) = 1 by CARD_1:30;
    hence t1 is context of x by CONTEXT;
  end;
