reserve i,j for Nat;
reserve i,j for Nat,
  x for variable,
  l for quasi-loci;
reserve C for initialized ConstructorSignature,
  c for constructor OperSymbol of C;
reserve a,a9 for quasi-adjective,
  t,t1,t2 for quasi-term,
  T for quasi-type,

  c for Element of Constructors;

theorem
 for T1,T2,T3 being quasi-type of C st T1 matches_with T2 & T2 matches_with T3
  holds T1 matches_with T3
  proof
    let t1,t2,t3 be quasi-type of C;
    given f1 being valuation of C such that
A1: (adjs t2) at f1 c= adjs t1 & (the_base_of t2) at f1 = the_base_of t1;
    given f2 being valuation of C such that
A2: (adjs t3) at f2 c= adjs t2 & (the_base_of t3) at f2 = the_base_of t2;
    take f2 at f1;
    ((adjs t3) at f2) at f1 c= (adjs t2) at f1 by A2,ABCMIZ_1:146; then
    ((adjs t3) at f2) at f1 c= adjs t1 by A1;
    hence thesis by A1,A2,ABCMIZ_1:149,150;
  end;
