reserve i for Nat, x,y for set;
reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;

theorem Th48:
  for S being (4,1) integer non empty non void BoolSignature
  for I being integer SortSymbol of S holds
  I <> the bool-sort of S &
  (the connectives of S).4 is_of_type {},I &
  (the connectives of S).(4+1) is_of_type {},I &
  (the connectives of S).4 <> (the connectives of S).(4+1) &
  (the connectives of S).(4+2) is_of_type <*I*>,I &
  (the connectives of S).(4+3) is_of_type <*I,I*>,I &
  (the connectives of S).(4+4) is_of_type <*I,I*>,I &
  (the connectives of S).(4+5) is_of_type <*I,I*>,I &
  (the connectives of S).(4+3) <> (the connectives of S).(4+4) &
  (the connectives of S).(4+3) <> (the connectives of S).(4+5) &
  (the connectives of S).(4+4) <> (the connectives of S).(4+5) &
  (the connectives of S).(4+6) is_of_type <*I,I*>,the bool-sort of S
  proof
    let S be (4,1) integer non empty non void BoolSignature;
    let I be integer SortSymbol of S;
A1: I = 1 by Def39;
    ex I being SortSymbol of S st I = 1 &
    I <> the bool-sort of S &
    (the connectives of S).4 is_of_type {},I &
    (the connectives of S).(4+1) is_of_type {},I &
    (the connectives of S).4 <> (the connectives of S).(4+1) &
    (the connectives of S).(4+2) is_of_type <*I*>,I &
    (the connectives of S).(4+3) is_of_type <*I,I*>,I &
    (the connectives of S).(4+4) is_of_type <*I,I*>,I &
    (the connectives of S).(4+5) is_of_type <*I,I*>,I &
    (the connectives of S).(4+3) <> (the connectives of S).(4+4) &
    (the connectives of S).(4+3) <> (the connectives of S).(4+5) &
    (the connectives of S).(4+4) <> (the connectives of S).(4+5) &
    (the connectives of S).(4+6) is_of_type <*I,I*>,the bool-sort of S by Def38
;
    hence thesis by A1;
  end;
