theorem Th55:
  len p = len the_arity_of c implies c-trm p <> (ast C)term(a,t)
proof
  assume len p = len the_arity_of c;
  then c-trm p = [c, the carrier of C]-tree p by Def35;
  then
A1: (c-trm p).{} = [c, the carrier of C] by TREES_4:def 4;
  assume c-trm p = (ast C)term(a,t);
  then c-trm p = [ *, the carrier of C]-tree<*a,t*> by Th46;
  then [c, the carrier of C] = [ *, the carrier of C] by A1,TREES_4:def 4;
  then c = * by XTUPLE_0:1;
  hence thesis by Def11;
end;
