reserve i for Nat,
  j for Element of NAT,
  X,Y,x,y,z for set;
reserve C for initialized ConstructorSignature,
  s for SortSymbol of C,
  o for OperSymbol of C,
  c for constructor OperSymbol of C;
reserve a,b for expression of C, an_Adj C;
reserve t, t1,t2 for expression of C, a_Type C;
reserve p for FinSequence of QuasiTerms C;

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;
