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;

theorem
  QuasiTerms C misses QuasiAdjs C & QuasiTerms C misses QuasiTypes C &
  QuasiTypes C misses QuasiAdjs C
proof
  set X = MSVars C;
  set Y = X (\/) ((the carrier of C) --> {0});
  ex A being MSSubset of FreeMSA Y st ( Free(C, X) = GenMSAlg
  A)&( A = (Reverse Y)""X) by MSAFREE3:def 1;
  then the Sorts of Free(C, X) is MSSubset of FreeMSA Y by MSUALG_2:def 9;
  then
A1: the Sorts of Free(C, X) c= the Sorts of FreeMSA Y by PBOOLE:def 18;
  then
A2: QuasiTerms C c= (the Sorts of FreeMSA Y).a_Term C;
A3: (the Sorts of Free(C,X)).an_Adj C c= (the Sorts of FreeMSA Y).an_Adj C
  by A1;
  QuasiAdjs C c= (the Sorts of Free(C,X)).an_Adj C
  proof
    let x be object;
    assume x in QuasiAdjs C;
    then ex a st x = a & a is regular;
    hence thesis by Def28;
  end;
  then
A4: QuasiAdjs C c= (the Sorts of FreeMSA Y).an_Adj C by A3;
  (the Sorts of FreeMSA Y).a_Term C misses (the Sorts of FreeMSA Y).an_Adj C
  by PROB_2:def 2;
  hence QuasiTerms C misses QuasiAdjs C by A2,A4,XBOOLE_1:64;
  now
    let x be object;
    assume that
A5: x in QuasiTerms C and
A6: x in QuasiTypes C;
    x is quasi-type of C by A6,Def43;
    hence contradiction by A5;
  end;
  hence QuasiTerms C misses QuasiTypes C by XBOOLE_0:3;
  now
    let x be object;
    assume that
A7: x in QuasiAdjs C and
A8: x in QuasiTypes C;
    x is quasi-type of C by A8,Def43;
    hence contradiction by A7;
  end;
  hence thesis by XBOOLE_0:3;
end;
