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;
reserve e for expression of C;
reserve a,a9 for expression of C, an_Adj C;
reserve q for pure expression of C, a_Type C,
  A for finite Subset of QuasiAdjs C;
reserve T for quasi-type of C;
reserve f for valuation of C;
reserve x for variable;

theorem Th150:
  for A being Subset of QuasiAdjs C for f1,f2 being valuation of C
  holds (A at f1) at f2 = A at (f1 at f2)
proof
  let A be Subset of QuasiAdjs C;
  let f1,f2 be valuation of C;
  thus (A at f1) at f2 c= A at (f1 at f2)
  proof
    let x be object;
    assume x in (A at f1) at f2;
    then consider a being quasi-adjective of C such that
A1: x = a at f2 and
A2: a in A at f1;
    consider b being quasi-adjective of C such that
A3: a = b at f1 and
A4: b in A by A2;
    x = b at (f1 at f2) by A1,A3,Th149;
    hence thesis by A4;
  end;
  let x be object;
  assume x in A at (f1 at f2);
  then consider a being quasi-adjective of C such that
A5: x = a at (f1 at f2) and
A6: a in A;
A7: x = a at f1 at f2 by A5,Th149;
  a at f1 in A at f1 by A6;
  hence thesis by A7;
end;
