theorem Th132:
  for xi being Node of t st t.xi = [x,s] holds t|xi = x-term
  proof
    let xi be Node of t;
    assume Z0: t.xi = [x,s];
    reconsider tx = t|xi as Element of Free(S,X) by MSAFREE4:44;
    per cases by Th16;
    suppose ex s1,x11 st tx = x11-term;
      then consider s1,x11 such that
A1:   tx = x11-term;
      <*>NAT in dom tx = (dom t)|xi by TREES_1:22,TREES_2:def 10;
      then tx.{} = t.(xi^{}) by TREES_2:def 10;
      hence t|xi = x-term by Z0,A1,TREES_4:3;
    end;
    suppose ex o,p st tx = o-term p;
      then consider o,p such that
A2:   tx = o-term p;
      <*>NAT in dom tx = (dom t)|xi by TREES_1:22,TREES_2:def 10;
      then tx.{} = t.(xi^{}) by TREES_2:def 10;
      then [o,the carrier of S] = [x,s] by Z0,A2,TREES_4:def 4;
      then s in the carrier of S = s by XTUPLE_0:1;
      hence thesis;
    end;
  end;
