theorem Th54:
  len p = len the_arity_of c implies c-trm p <> (non_op C)term a
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 = (non_op C)term a;
  then c-trm p = [non_op, the carrier of C]-tree<*a*> by Th43;
  then [c, the carrier of C] = [non_op, the carrier of C] by A1,TREES_4:def 4;
  then c = non_op by XTUPLE_0:1;
  hence thesis by Def11;
end;
