reserve i,j for Nat;

theorem
 for C being initialized standardized ConstructorSignature
 for t being expression of C holds
   t is quasi-term of C iff
      (t.{})`1 in Constructors & (t.{})`1`1 = a_Term or (t.{})`1 in Vars
  proof let C be initialized standardized ConstructorSignature;
   let t be expression of C;
   hereby assume t is quasi-term of C; then
     reconsider tr = t as quasi-term of C;
     tr is compound or tr is non compound;
     hence (t.{})`1 in Constructors & (t.{})`1`1 = a_Term or
           (t.{})`1 in Vars by Th21,Th22;
    end;
   assume that
A1: (t.{})`1 in Constructors & (t.{})`1`1 = a_Term or (t.{})`1 in Vars and
A2: t is not quasi-term of C;
A3: (t.{})`1 in Vars implies
      ex x being Element of Vars st x = (t.{})`1 & t = x-term C by Th17; then
    (t.{})`1 in Constructors & (t.{})`1`1 = a_Term by A1,A2; then
    ex o being OperSymbol of C st o = (t.{})`1 & the_result_sort_of o = o`1 &
     t is expression of C, the_result_sort_of o by Th20;
   hence thesis by A2,A3,A1;
  end;
