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 Th145:
  for A,B being Subset of QuasiAdjs C
  holds (A \/ B) at f = (A at f) \/ (B at f)
proof
  let A,B be Subset of QuasiAdjs C;
  thus (A \/ B) at f c= (A at f) \/ (B at f)
  proof
    let x be object;
    assume x in (A \/ B) at f;
    then consider a being quasi-adjective of C such that
A1: x = a at f and
A2: a in A \/ B;
    a in A or a in B by A2,XBOOLE_0:def 3;
    then x in A at f or x in B at f by A1;
    hence thesis by XBOOLE_0:def 3;
  end;
  let x be object;
  assume x in (A at f) \/ (B at f);
  then x in (A at f) or x in (B at f) by XBOOLE_0:def 3;
  then
  A c= A\/B & (ex a being quasi-adjective of C st x = a at f & a in A) or
  B c= A\/B & ex a being quasi-adjective of C st x = a at f & a in B
  by XBOOLE_1:7;
  hence thesis;
end;
