theorem Lem13:
  t.a = [x,s] implies a in Leaves dom t
  proof
    assume Z0: t.a = [x,s];
    then reconsider q = a as Element of dom t by FUNCT_1:def 2;
    reconsider v = t|q as Element of Free(S,X) by MSAFREE4:44;
    {} in (dom t)|q by TREES_1:22;
    then
A2: v.{} = t.(q^<*>NAT) by TREES_2:def 10;
    per cases by Th16;
    suppose ex o,p st v = o-term p;
      then consider o,p such that
A1:   v = o-term p;
      [o,the carrier of S] = v.{} = [x,s] by Z0,A1,A2,TREES_4:def 4;
      then s in the carrier of S = s by XTUPLE_0:1;
      hence a in Leaves dom t;
    end;
    suppose ex s1,x11 st v = x11-term;
      then consider s1,x11 such that
A1:   v = x11-term;
      reconsider r = <*0*> as FinSequence of NAT;
      now assume q^<*0*> in dom t;
        then r in (dom t)|q = dom v by TREES_1:def 6,TREES_2:def 10;
        then <*0*> in {{}} by A1,TREES_1:29;
        hence contradiction;
      end;
      hence a in Leaves dom t by TREES_1:54;
    end;
  end;
