:: VALUAT_1 semantic presentation

begin

definition
let Al be ( ( ) ( Relation-like non empty ) QC-alphabet ) ;
let A be ( ( ) ( ) set ) ;
func Valuations_in (Al,A) -> ( ( ) ( ) set ) equals :: VALUAT_1:def 1
 Funcs ((bound_QC-variables Al : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ext-real ) ( ext-real ) set ) ) : ( ( ) ( functional ) set ) ;
end;

registration
let Al be ( ( ) ( Relation-like non empty ) QC-alphabet ) ;
let A be ( ( non empty ) ( non empty ) set ) ;
cluster Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) -> functional non empty ;
end;

theorem :: VALUAT_1:1
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) is ( ( ) ( Relation-like Function-like ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) set ) ) holds
x : ( ( ) ( ) set ) is ( ( Function-like quasi_total ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Function of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ;

definition
let Al be ( ( ) ( Relation-like non empty ) QC-alphabet ) ;
let A be ( ( non empty ) ( non empty ) set ) ;
:: original: Valuations_in
redefine func Valuations_in (Al,A) -> ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ;
end;

definition
let Al be ( ( ) ( Relation-like non empty ) QC-alphabet ) ;
let A be ( ( non empty ) ( non empty ) set ) ;
let x be ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;
let p be ( ( ) ( Relation-like Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) ;
func FOR_ALL (x,p) -> ( ( ) ( Relation-like Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) means :: VALUAT_1:def 2
 for v being ( ( ) ( Relation-like bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( ) ( ) set ) -valued Function-like V24( bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ) holds it : ( ( Relation-like A : ( ( ) ( ) set ) -valued Function-like ) ( Relation-like A : ( ( ) ( ) set ) -valued Function-like ) set ) . v : ( ( ) ( Relation-like bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = ALL { (p : ( ( ) ( ) set ) . v9 : ( ( ) ( Relation-like bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) where v9 is ( ( ) ( Relation-like bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( ) ( ) set ) -valued Function-like V24( bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ) : for y being ( ( ) ( ) bound_QC-variable of ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) st x : ( ( ) ( ) set ) <> y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) holds
v9 : ( ( ) ( Relation-like bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ) . y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of A : ( ( ) ( ) set ) ) = v : ( ( ) ( Relation-like bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ) . y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of A : ( ( ) ( ) set ) ) } : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) ;
end;

theorem :: VALUAT_1:2
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for x being ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )
for p being ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) holds
( (FOR_ALL (x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,p : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = FALSE : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) iff ex v1 being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) st
( p : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v1 : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = FALSE : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) & ( for y being ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) st x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) <> y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) holds
v1 : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) . y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) = v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) . y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) ) ) ) ;

theorem :: VALUAT_1:3
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for x being ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )
for p being ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) holds
( (FOR_ALL (x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,p : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = TRUE : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) iff for v1 being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) st ( for y being ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) st x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) <> y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) holds
v1 : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) . y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) = v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) . y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) ) holds
p : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v1 : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = TRUE : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) ) ;

notation
let Al be ( ( ) ( Relation-like non empty ) QC-alphabet ) ;
let A be ( ( non empty ) ( non empty ) set ) ;
let k be ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let ll be ( ( bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued V47(k : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -valued bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V47(k : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) CQC-variable_list of k : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty ) set ) ) ;
let v be ( ( ) ( Relation-like bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ) ;
synonym v *' ll for A * Al;
end;

definition
let Al be ( ( ) ( Relation-like non empty ) QC-alphabet ) ;
let A be ( ( non empty ) ( non empty ) set ) ;
let k be ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let ll be ( ( bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued V47(k : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -valued bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V47(k : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) CQC-variable_list of k : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty ) set ) ) ;
let v be ( ( ) ( Relation-like bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ) ;
:: original: *'
redefine func v *' ll -> ( ( ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( ) ( ) set ) -valued Function-like FinSequence-like ) FinSequence of A : ( ( ) ( ) set ) ) means :: VALUAT_1:def 3
( len it : ( ( Function-like quasi_total ) ( Relation-like k : ( ( ) ( ) set ) -defined ll : ( ( ) ( ) set ) -valued Function-like V24(k : ( ( ) ( ) set ) ) quasi_total ) Element of K24(K25(k : ( ( ) ( ) set ) ,ll : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) = k : ( ( ) ( ) set ) & ( for i being ( ( natural ) ( ext-real ordinal natural V40() V45() ) number ) st 1 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() non empty ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( natural ) ( ext-real ordinal natural V40() V45() ) number ) & i : ( ( natural ) ( ext-real ordinal natural V40() V45() ) number ) <= k : ( ( ) ( ) set ) holds
it : ( ( Function-like quasi_total ) ( Relation-like k : ( ( ) ( ) set ) -defined ll : ( ( ) ( ) set ) -valued Function-like V24(k : ( ( ) ( ) set ) ) quasi_total ) Element of K24(K25(k : ( ( ) ( ) set ) ,ll : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . i : ( ( natural ) ( ext-real ordinal natural V40() V45() ) number ) : ( ( ) ( ) set ) = v : ( ( Relation-like A : ( ( ) ( ) set ) -valued Function-like ) ( Relation-like A : ( ( ) ( ) set ) -valued Function-like ) set ) . (ll : ( ( ) ( ) set ) . i : ( ( natural ) ( ext-real ordinal natural V40() V45() ) number ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) );
end;

definition
let Al be ( ( ) ( Relation-like non empty ) QC-alphabet ) ;
let A be ( ( non empty ) ( non empty ) set ) ;
let k be ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let ll be ( ( bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued V47(k : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -valued bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V47(k : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) CQC-variable_list of k : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty ) set ) ) ;
let r be ( ( ) ( ) Element of relations_on A : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;
func ll 'in' r -> ( ( ) ( Relation-like Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) means :: VALUAT_1:def 4
 for v being ( ( ) ( Relation-like bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( ) ( ) set ) -valued Function-like V24( bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ) holds
( ( v : ( ( ) ( Relation-like bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ) *' ll : ( ( ) ( ) set ) : ( ( ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( ) ( ) set ) -valued Function-like FinSequence-like ) FinSequence of A : ( ( ) ( ) set ) ) in r : ( ( Relation-like A : ( ( ) ( ) set ) -valued Function-like ) ( Relation-like A : ( ( ) ( ) set ) -valued Function-like ) set ) implies it : ( ( Function-like quasi_total ) ( Relation-like k : ( ( ) ( ) set ) -defined ll : ( ( ) ( ) set ) -valued Function-like V24(k : ( ( ) ( ) set ) ) quasi_total ) Element of K24(K25(k : ( ( ) ( ) set ) ,ll : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . v : ( ( ) ( Relation-like bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = TRUE : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) ) & ( it : ( ( Function-like quasi_total ) ( Relation-like k : ( ( ) ( ) set ) -defined ll : ( ( ) ( ) set ) -valued Function-like V24(k : ( ( ) ( ) set ) ) quasi_total ) Element of K24(K25(k : ( ( ) ( ) set ) ,ll : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . v : ( ( ) ( Relation-like bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = TRUE : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) implies v : ( ( ) ( Relation-like bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ) *' ll : ( ( ) ( ) set ) : ( ( ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( ) ( ) set ) -valued Function-like FinSequence-like ) FinSequence of A : ( ( ) ( ) set ) ) in r : ( ( Relation-like A : ( ( ) ( ) set ) -valued Function-like ) ( Relation-like A : ( ( ) ( ) set ) -valued Function-like ) set ) ) & ( not v : ( ( ) ( Relation-like bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ) *' ll : ( ( ) ( ) set ) : ( ( ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( ) ( ) set ) -valued Function-like FinSequence-like ) FinSequence of A : ( ( ) ( ) set ) ) in r : ( ( Relation-like A : ( ( ) ( ) set ) -valued Function-like ) ( Relation-like A : ( ( ) ( ) set ) -valued Function-like ) set ) implies it : ( ( Function-like quasi_total ) ( Relation-like k : ( ( ) ( ) set ) -defined ll : ( ( ) ( ) set ) -valued Function-like V24(k : ( ( ) ( ) set ) ) quasi_total ) Element of K24(K25(k : ( ( ) ( ) set ) ,ll : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . v : ( ( ) ( Relation-like bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = FALSE : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) ) & ( it : ( ( Function-like quasi_total ) ( Relation-like k : ( ( ) ( ) set ) -defined ll : ( ( ) ( ) set ) -valued Function-like V24(k : ( ( ) ( ) set ) ) quasi_total ) Element of K24(K25(k : ( ( ) ( ) set ) ,ll : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . v : ( ( ) ( Relation-like bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = FALSE : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) implies not v : ( ( ) ( Relation-like bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ) *' ll : ( ( ) ( ) set ) : ( ( ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( ) ( ) set ) -valued Function-like FinSequence-like ) FinSequence of A : ( ( ) ( ) set ) ) in r : ( ( Relation-like A : ( ( ) ( ) set ) -valued Function-like ) ( Relation-like A : ( ( ) ( ) set ) -valued Function-like ) set ) ) );
end;

definition
let Al be ( ( ) ( Relation-like non empty ) QC-alphabet ) ;
let A be ( ( non empty ) ( non empty ) set ) ;
let F be ( ( Function-like quasi_total ) ( Relation-like CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V24( CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Function of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) , Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) ;
let p be ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;
:: original: .
redefine func F . p -> ( ( ) ( Relation-like Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) ;
end;

definition
let Al be ( ( ) ( Relation-like non empty ) QC-alphabet ) ;
let D be ( ( non empty ) ( non empty ) set ) ;
mode interpretation of Al,D -> ( ( Function-like quasi_total ) ( Relation-like QC-pred_symbols Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -defined relations_on D : ( ( ) ( ) set ) : ( ( ) ( ) set ) -valued Function-like non empty V24( QC-pred_symbols Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) quasi_total ) Function of QC-pred_symbols Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) , relations_on D : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) means :: VALUAT_1:def 5
 for P being ( ( ) ( ) Element of QC-pred_symbols Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) )
for r being ( ( ) ( ) Element of relations_on D : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) holds
( not it : ( ( ) ( ) set ) . P : ( ( ) ( ) Element of QC-pred_symbols Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of relations_on D : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) = r : ( ( ) ( ) Element of relations_on D : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) or r : ( ( ) ( ) Element of relations_on D : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = empty_rel D : ( ( ) ( ) set ) : ( ( ) ( ) Element of relations_on D : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) or the_arity_of P : ( ( ) ( ) Element of QC-pred_symbols Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) = the_arity_of r : ( ( ) ( ) Element of relations_on D : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) );
end;

definition
let Al be ( ( ) ( Relation-like non empty ) QC-alphabet ) ;
let A be ( ( non empty ) ( non empty ) set ) ;
let k be ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let J be ( ( ) ( Relation-like QC-pred_symbols Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on A : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) ;
let P be ( ( ) ( ) QC-pred_symbol of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) , ( ( ) ( non empty ) Element of K24((QC-pred_symbols Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;
:: original: .
redefine func J . P -> ( ( ) ( ) Element of relations_on A : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ;
end;

definition
let Al be ( ( ) ( Relation-like non empty ) QC-alphabet ) ;
let A be ( ( non empty ) ( non empty ) set ) ;
let J be ( ( ) ( Relation-like QC-pred_symbols Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on A : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) ;
let p be ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;
func Valid (p,J) -> ( ( ) ( Relation-like Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) means :: VALUAT_1:def 6
 ex F being ( ( Function-like quasi_total ) ( Relation-like CQC-WFF Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-WFF Al : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined Funcs ((Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V24( CQC-WFF Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-WFF Al : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Function of CQC-WFF Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-WFF Al : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) , Funcs ((Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) st
( it : ( ( Relation-like A : ( ( ) ( ) set ) -valued Function-like ) ( Relation-like A : ( ( ) ( ) set ) -valued Function-like ) set ) = F : ( ( Function-like quasi_total ) ( Relation-like CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V24( CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Function of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) , Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . p : ( ( ) ( ) set ) : ( ( ) ( Relation-like Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) & F : ( ( Function-like quasi_total ) ( Relation-like CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V24( CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Function of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) , Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . (VERUM Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of CQC-WFF Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-WFF Al : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) = (Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) --> TRUE : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( Relation-like Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ) quasi_total boolean-valued ) Element of K24(K25((Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & ( for p, q being ( ( ) ( ) Element of CQC-WFF Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-WFF Al : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for x being ( ( ) ( ) bound_QC-variable of ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )
for k being ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) )
for ll being ( ( bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued V47(b5 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V47(b5 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) CQC-variable_list of k : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) , ( ( ) ( ) set ) )
for P being ( ( ) ( ) QC-pred_symbol of Al : ( ( non empty ) ( non empty ) set ) , ( ( ) ( ) Element of K24((QC-pred_symbols Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) holds
( F : ( ( Function-like quasi_total ) ( Relation-like CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V24( CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Function of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) , Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . (P : ( ( ) ( ) QC-pred_symbol of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) , ( ( ) ( non empty ) Element of K24((QC-pred_symbols Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ! ll : ( ( bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued V47(b5 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -valued bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V47(b5 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) CQC-variable_list of b5 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of CQC-WFF Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-WFF Al : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) = ll : ( ( bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued V47(b5 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -valued bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V47(b5 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) CQC-variable_list of b5 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty ) set ) ) 'in' (J : ( ( ) ( ) set ) . P : ( ( ) ( ) QC-pred_symbol of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) , ( ( ) ( non empty ) Element of K24((QC-pred_symbols Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of relations_on A : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) & F : ( ( Function-like quasi_total ) ( Relation-like CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V24( CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Function of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) , Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . ('not' p : ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-WFF Al : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) = 'not' (F : ( ( Function-like quasi_total ) ( Relation-like CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V24( CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Function of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) , Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . p : ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ) quasi_total boolean-valued ) Element of K24(K25((Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & F : ( ( Function-like quasi_total ) ( Relation-like CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V24( CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Function of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) , Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . (p : ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) '&' q : ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-WFF Al : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) = (F : ( ( Function-like quasi_total ) ( Relation-like CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V24( CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Function of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) , Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . p : ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) '&' (F : ( ( Function-like quasi_total ) ( Relation-like CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V24( CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Function of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) , Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . q : ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ) quasi_total boolean-valued ) Element of K24(K25((Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & F : ( ( Function-like quasi_total ) ( Relation-like CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V24( CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Function of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) , Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . (All (x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,p : ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( ) Element of CQC-WFF Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-WFF Al : ( ( non empty ) ( non empty ) set ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) = FOR_ALL (x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,(F : ( ( Function-like quasi_total ) ( Relation-like CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) -valued Function-like non empty V24( CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Function of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) , Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . p : ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) ) ) );
end;

theorem :: VALUAT_1:4
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds Valid ((VERUM Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) = (Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) --> TRUE : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total boolean-valued ) Element of K24(K25((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: VALUAT_1:5
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds (Valid ((VERUM Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = TRUE : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) ;

theorem :: VALUAT_1:6
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for k being ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) )
for A being ( ( non empty ) ( non empty ) set )
for ll being ( ( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued V47(b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -valued bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V47(b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) CQC-variable_list of k : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty ) set ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b3 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( ) QC-pred_symbol of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) , ( ( ) ( non empty ) Element of K24((QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) holds Valid ((P : ( ( ) ( ) QC-pred_symbol of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) , ( ( ) ( non empty ) Element of K24((QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ! ll : ( ( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued V47(b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -valued bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V47(b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) CQC-variable_list of b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b3 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) = ll : ( ( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued V47(b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -valued bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V47(b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) CQC-variable_list of b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty ) set ) ) 'in' (J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b3 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) . P : ( ( ) ( ) QC-pred_symbol of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) , ( ( ) ( non empty ) Element of K24((QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of relations_on b3 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) ;

theorem :: VALUAT_1:7
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for k being ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) )
for A being ( ( non empty ) ( non empty ) set )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) )
for ll being ( ( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued V47(b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -valued bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V47(b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) CQC-variable_list of k : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty ) set ) )
for p being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b3 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( ) QC-pred_symbol of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) , ( ( ) ( non empty ) Element of K24((QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( ) ( ) Element of relations_on A : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) st p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) = P : ( ( ) ( ) QC-pred_symbol of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) , ( ( ) ( non empty ) Element of K24((QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ! ll : ( ( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued V47(b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -valued bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V47(b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) CQC-variable_list of b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) & r : ( ( ) ( ) Element of relations_on b3 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b3 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) . P : ( ( ) ( ) QC-pred_symbol of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) , ( ( ) ( non empty ) Element of K24((QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of relations_on b3 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
( v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) *' ll : ( ( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued V47(b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -valued bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V47(b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) CQC-variable_list of b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like FinSequence-like ) FinSequence of b3 : ( ( non empty ) ( non empty ) set ) ) in r : ( ( ) ( ) Element of relations_on b3 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) iff (Valid (p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b3 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = TRUE : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) ) ;

theorem :: VALUAT_1:8
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for k being ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) )
for A being ( ( non empty ) ( non empty ) set )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) )
for ll being ( ( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued V47(b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -valued bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V47(b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) CQC-variable_list of k : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty ) set ) )
for p being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b3 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( ) QC-pred_symbol of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) , ( ( ) ( non empty ) Element of K24((QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( ) ( ) Element of relations_on A : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) st p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) = P : ( ( ) ( ) QC-pred_symbol of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) , ( ( ) ( non empty ) Element of K24((QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ! ll : ( ( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued V47(b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -valued bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V47(b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) CQC-variable_list of b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) & r : ( ( ) ( ) Element of relations_on b3 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b3 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) . P : ( ( ) ( ) QC-pred_symbol of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) , ( ( ) ( non empty ) Element of K24((QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of relations_on b3 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
( not v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) *' ll : ( ( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued V47(b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -valued bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V47(b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) CQC-variable_list of b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like FinSequence-like ) FinSequence of b3 : ( ( non empty ) ( non empty ) set ) ) in r : ( ( ) ( ) Element of relations_on b3 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) iff (Valid (p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b3 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = FALSE : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) ) ;

theorem :: VALUAT_1:9
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for p being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds Valid (('not' p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) = 'not' (Valid (p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total boolean-valued ) Element of K24(K25((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: VALUAT_1:10
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )
for p being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds (Valid (('not' p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = 'not' ((Valid (p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) ;

theorem :: VALUAT_1:11
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for p, q being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds Valid ((p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) '&' q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) = (Valid (p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) '&' (Valid (q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total boolean-valued ) Element of K24(K25((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: VALUAT_1:12
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )
for p, q being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds (Valid ((p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) '&' q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = ((Valid (p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) '&' ((Valid (q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) ;

theorem :: VALUAT_1:13
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for x being ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for p being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds Valid ((All (x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) = FOR_ALL (x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,(Valid (p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) ;

theorem :: VALUAT_1:14
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )
for p being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds (Valid ((p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) '&' ('not' p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = FALSE : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) ;

theorem :: VALUAT_1:15
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )
for p being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds (Valid (('not' (p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) '&' ('not' p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = TRUE : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) ;

definition
let Al be ( ( ) ( Relation-like non empty ) QC-alphabet ) ;
let A be ( ( non empty ) ( non empty ) set ) ;
let p be ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;
let J be ( ( ) ( Relation-like QC-pred_symbols Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on A : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) ;
let v be ( ( ) ( Relation-like bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ) ;
pred J,v |= p means :: VALUAT_1:def 7
(Valid (p : ( ( ) ( ) set ) ,J : ( ( ) ( ) set ) )) : ( ( ) ( Relation-like Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v : ( ( Relation-like A : ( ( ) ( ) set ) -valued Function-like ) ( Relation-like A : ( ( ) ( ) set ) -valued Function-like ) set ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = TRUE : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) ;
end;

theorem :: VALUAT_1:16
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for k being ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) )
for A being ( ( non empty ) ( non empty ) set )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) )
for ll being ( ( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued V47(b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -valued bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V47(b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) CQC-variable_list of k : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty ) set ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b3 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )
for P being ( ( ) ( ) QC-pred_symbol of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) , ( ( ) ( non empty ) Element of K24((QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) holds
( J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b3 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ,v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) |= P : ( ( ) ( ) QC-pred_symbol of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) , ( ( ) ( non empty ) Element of K24((QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ! ll : ( ( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued V47(b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -valued bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V47(b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) CQC-variable_list of b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) iff (ll : ( ( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued V47(b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ( Relation-like NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) -defined QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -valued bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V47(b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) CQC-variable_list of b2 : ( ( ) ( ext-real V4() V5() V6() V7() V8() V9() ordinal natural V40() V45() ) Element of NAT : ( ( ) ( V4() V5() V6() V7() V8() V9() V10() ordinal V40() V45() V46() ) Element of K24(REAL : ( ( ) ( V4() V5() V6() V10() ) set ) ) : ( ( ) ( ) set ) ) ) , ( ( ) ( non empty ) set ) ) 'in' (J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b3 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) . P : ( ( ) ( ) QC-pred_symbol of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) , ( ( ) ( non empty ) Element of K24((QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of relations_on b3 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = TRUE : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) ) ;

theorem :: VALUAT_1:17
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )
for p being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds
( J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) |= 'not' p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) iff not J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) |= p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: VALUAT_1:18
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )
for p, q being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds
( J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) |= p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) '&' q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) iff ( J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) |= p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) & J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) |= q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ;

theorem :: VALUAT_1:19
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for x being ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )
for p being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds
( J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) |= All (x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) iff (FOR_ALL (x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,(Valid (p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = TRUE : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) ) ;

theorem :: VALUAT_1:20
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for x being ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )
for p being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds
( J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) |= All (x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) iff for v1 being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) st ( for y being ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) st x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) <> y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) holds
v1 : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) . y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) = v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) . y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) ) holds
(Valid (p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v1 : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = TRUE : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) ) ;

theorem :: VALUAT_1:21
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for p being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds Valid (('not' ('not' p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) = Valid (p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) ;

theorem :: VALUAT_1:22
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for p being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds Valid ((p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) '&' p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) = Valid (p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) ;

theorem :: VALUAT_1:23
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )
for p, q being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds
( J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) |= p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) => q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) iff ( (Valid (p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = FALSE : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) or (Valid (q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = TRUE : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: VALUAT_1:24
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )
for p, q being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds
( J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) |= p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) => q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) iff ( J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) |= p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) implies J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) |= q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ;

theorem :: VALUAT_1:25
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for x being ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )
for p being ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) st (FOR_ALL (x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,p : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = TRUE : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) holds
p : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = TRUE : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) ;

definition
let Al be ( ( ) ( Relation-like non empty ) QC-alphabet ) ;
let A be ( ( non empty ) ( non empty ) set ) ;
let J be ( ( ) ( Relation-like QC-pred_symbols Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on A : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) ;
let p be ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;
pred J |= p means :: VALUAT_1:def 8
 for v being ( ( ) ( Relation-like bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( ) ( ) set ) -valued Function-like V24( bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( non empty ) ( non empty ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) Element of K24((QC-variables Al : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( ) ( ) set ) ) ) holds J : ( ( ) ( ) set ) ,v : ( ( ) ( Relation-like bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined A : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty ) ( non empty ) set ) ) ) |= p : ( ( ) ( ) set ) ;
end;

theorem :: VALUAT_1:26
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for Y, Z being ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for V1, V2 being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ex v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) st
( ( for x being ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) st x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) <> Y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) holds
v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) . x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) = V1 : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) . x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) ) & v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) . Y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) = V2 : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) . Z : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: VALUAT_1:27
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for x being ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for p being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) st not x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) in still_not-bound_in p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K24((bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) holds
for v, w being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) st ( for y being ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) st x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) <> y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) holds
w : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) . y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) = v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) . y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) ) holds
(Valid (p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = (Valid (p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . w : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) ;

theorem :: VALUAT_1:28
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for x being ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )
for p being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) st J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) |= p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) & not x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) in still_not-bound_in p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K24((bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) holds
for w being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) st ( for y being ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) st x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) <> y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) holds
w : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) . y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) = v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) . y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) ) holds
J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,w : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) |= p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: VALUAT_1:29
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for x being ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )
for p being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds
( J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) |= All (x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) iff for w being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) st ( for y being ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) st x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) <> y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) holds
w : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) . y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) = v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) . y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) ) holds
J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,w : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) |= p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: VALUAT_1:30
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for x, y being ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for p, q being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )
for s9 being ( ( ) ( ) QC-formula of ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) <> y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) & p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) = s9 : ( ( ) ( ) QC-formula of ( ( non empty ) ( non empty ) set ) ) . x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( non empty ) ( non empty ) set ) ) & q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) = s9 : ( ( ) ( ) QC-formula of ( ( non empty ) ( non empty ) set ) ) . y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( non empty ) ( non empty ) set ) ) holds
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) st v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) . x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) = v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) . y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) holds
(Valid (p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) = (Valid (q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( Relation-like Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined BOOLEAN : ( ( ) ( non empty ) set ) -valued Function-like V24( Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) quasi_total ) Element of Funcs ((Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,BOOLEAN : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , BOOLEAN : ( ( ) ( non empty ) set ) ) ) . v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( boolean ) Element of BOOLEAN : ( ( ) ( non empty ) set ) ) ;

theorem :: VALUAT_1:31
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for x, y being ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for s9 being ( ( ) ( ) QC-formula of ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) <> y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) & not x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) in still_not-bound_in s9 : ( ( ) ( ) QC-formula of ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of K24((bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) holds
not x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) in still_not-bound_in (s9 : ( ( ) ( ) QC-formula of ( ( non empty ) ( non empty ) set ) ) . y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of K24((bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ;

theorem :: VALUAT_1:32
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) |= VERUM Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: VALUAT_1:33
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )
for p, q being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) |= (p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) '&' q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) => (q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) '&' p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: VALUAT_1:34
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )
for p being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) |= (('not' p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) => p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) => p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: VALUAT_1:35
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )
for p, q being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) |= p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) => (('not' p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) => q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: VALUAT_1:36
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )
for p, q, t being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) |= (p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) => q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) => (('not' (q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) '&' t : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) => ('not' (p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) '&' t : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: VALUAT_1:37
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )
for p, q being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) st J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) |= p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) & J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) |= p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) => q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) holds
J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) |= q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: VALUAT_1:38
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for x being ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for v being ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )
for p being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,v : ( ( ) ( Relation-like bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V24( bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) quasi_total ) Element of Valuations_in (b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( functional non empty ) FUNCTION_DOMAIN of bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) |= (All (x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) => p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: VALUAT_1:39
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) |= VERUM Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: VALUAT_1:40
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for p, q being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) |= (p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) '&' q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) => (q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) '&' p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: VALUAT_1:41
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for p being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) |= (('not' p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) => p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) => p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: VALUAT_1:42
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for p, q being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) |= p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) => (('not' p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) => q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: VALUAT_1:43
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for p, q, t being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) |= (p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) => q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) => (('not' (q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) '&' t : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) => ('not' (p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) '&' t : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: VALUAT_1:44
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for p, q being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) st J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) |= p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) & J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) |= p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) => q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) holds
J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) |= q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: VALUAT_1:45
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for x being ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for p being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) holds J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) |= (All (x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) => p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: VALUAT_1:46
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for x being ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for p, q being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) ) st J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) |= p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) => q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) & not x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) in still_not-bound_in p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K24((bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) holds
J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) |= p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) => (All (x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ,q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: VALUAT_1:47
for Al being ( ( ) ( Relation-like non empty ) QC-alphabet )
for A being ( ( non empty ) ( non empty ) set )
for x, y being ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for p, q being ( ( ) ( ) Element of CQC-WFF Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) )
for J being ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of Al : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,A : ( ( non empty ) ( non empty ) set ) )
for s being ( ( ) ( ) QC-formula of ( ( non empty ) ( non empty ) set ) ) st p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) = s : ( ( ) ( ) QC-formula of ( ( non empty ) ( non empty ) set ) ) . x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( non empty ) ( non empty ) set ) ) & q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) = s : ( ( ) ( ) QC-formula of ( ( non empty ) ( non empty ) set ) ) . y : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( non empty ) ( non empty ) set ) ) & not x : ( ( ) ( ) bound_QC-variable of ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) in still_not-bound_in s : ( ( ) ( ) QC-formula of ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of K24((bound_QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) Element of K24((QC-variables b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) & J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) |= p : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) holds
J : ( ( ) ( Relation-like QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) -defined relations_on b2 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V24( QC-pred_symbols b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) set ) ) quasi_total ) interpretation of b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ,b2 : ( ( non empty ) ( non empty ) set ) ) |= q : ( ( ) ( ) Element of CQC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) : ( ( ) ( non empty ) Element of K24((QC-WFF b1 : ( ( ) ( Relation-like non empty ) QC-alphabet ) ) : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) ;