:: HILBERT3 semantic presentation

begin

registration
let m, n be ( ( non zero V13() ) ( ext-real non zero V7() V8() V9() V13() complex V15() integer ) Nat) ;
cluster (0 : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ,n : ( ( non zero V13() ) ( ext-real non zero V7() V8() V9() V13() complex V15() integer ) set ) ) --> (m : ( ( non zero V13() ) ( ext-real non zero V7() V8() V9() V13() complex V15() integer ) set ) ,0 : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like Function-like ) set ) -> one-to-one ;
cluster (n : ( ( non zero V13() ) ( ext-real non zero V7() V8() V9() V13() complex V15() integer ) set ) ,0 : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ) --> (0 : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ,m : ( ( non zero V13() ) ( ext-real non zero V7() V8() V9() V13() complex V15() integer ) set ) ) : ( ( ) ( Relation-like Function-like ) set ) -> one-to-one ;
end;

theorem :: HILBERT3:1
for i being ( ( integer ) ( ext-real complex V15() integer ) Integer) holds
( i : ( ( integer ) ( ext-real complex V15() integer ) Integer) is even iff i : ( ( integer ) ( ext-real complex V15() integer ) Integer) - 1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real complex V15() integer ) set ) is odd ) ;

theorem :: HILBERT3:2
for i being ( ( integer ) ( ext-real complex V15() integer ) Integer) holds
( i : ( ( integer ) ( ext-real complex V15() integer ) Integer) is odd iff i : ( ( integer ) ( ext-real complex V15() integer ) Integer) - 1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ext-real complex V15() integer ) set ) is even ) ;

theorem :: HILBERT3:3
for X being ( ( trivial ) ( trivial ) set )
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in X : ( ( trivial ) ( trivial ) set ) holds
for f being ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( trivial ) ( trivial ) set ) -defined b1 : ( ( trivial ) ( trivial ) set ) -valued Function-like total quasi_total ) Function of X : ( ( trivial ) ( trivial ) set ) ,X : ( ( trivial ) ( trivial ) set ) ) holds x : ( ( ) ( ) set ) is_a_fixpoint_of f : ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( trivial ) ( trivial ) set ) -defined b1 : ( ( trivial ) ( trivial ) set ) -valued Function-like total quasi_total ) Function of b1 : ( ( trivial ) ( trivial ) set ) ,b1 : ( ( trivial ) ( trivial ) set ) ) ;

theorem :: HILBERT3:4
for f being ( ( Relation-like Function-like Function-yielding ) ( Relation-like Function-like Function-yielding V55() ) Function) holds SubFuncs (rng f : ( ( Relation-like Function-like Function-yielding ) ( Relation-like Function-like Function-yielding V55() ) Function) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) = rng f : ( ( Relation-like Function-like Function-yielding ) ( Relation-like Function-like Function-yielding V55() ) Function) : ( ( ) ( ) set ) ;

theorem :: HILBERT3:5
for A, B, x being ( ( ) ( ) set )
for f being ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) st x : ( ( ) ( ) set ) in A : ( ( ) ( ) set ) & f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) in Funcs (A : ( ( ) ( ) set ) ,B : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) holds
f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . x : ( ( ) ( ) set ) : ( ( ) ( ) set ) in B : ( ( ) ( ) set ) ;

theorem :: HILBERT3:6
for A, B, C being ( ( ) ( ) set ) st ( not C : ( ( ) ( ) set ) = {} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) or B : ( ( ) ( ) set ) = {} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) or A : ( ( ) ( ) set ) = {} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) ) holds
for f being ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( ) ( ) set ) -defined Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) -valued Function-like quasi_total Function-yielding V55() ) Function of A : ( ( ) ( ) set ) , Funcs (B : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) ) holds doms f : ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( ) ( ) set ) -defined Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) -valued Function-like quasi_total Function-yielding V55() ) Function of b1 : ( ( ) ( ) set ) , Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) = A : ( ( ) ( ) set ) --> B : ( ( ) ( ) set ) : ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( ) ( ) set ) -defined {b2 : ( ( ) ( ) set ) } : ( ( ) ( non empty trivial ) set ) -valued Function-like constant total quasi_total ) Element of bool [:b1 : ( ( ) ( ) set ) ,{b2 : ( ( ) ( ) set ) } : ( ( ) ( non empty trivial ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: HILBERT3:7
for x being ( ( ) ( ) set ) holds {} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) . x : ( ( ) ( ) set ) : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) = {} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) ;

theorem :: HILBERT3:8
for X being ( ( ) ( ) set )
for A being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds ((0 : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ) --> (1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ,0 : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like {0 : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) -defined NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like one-to-one total quasi_total ) Element of bool [:{0 : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ,NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) * (chi (A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ,X : ( ( ) ( ) set ) )) : ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( ) ( ) set ) -defined {{} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) ,1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of bool [:b1 : ( ( ) ( ) set ) ,{{} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) ,1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total ) Element of bool [:b1 : ( ( ) ( ) set ) ,NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = chi ((A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) `) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ,X : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( ) ( ) set ) -defined {{} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) ,1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of bool [:b1 : ( ( ) ( ) set ) ,{{} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) ,1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: HILBERT3:9
for X being ( ( ) ( ) set )
for A being ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) holds ((0 : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ) --> (1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ,0 : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like {0 : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) -defined NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like one-to-one total quasi_total ) Element of bool [:{0 : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ,NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) * (chi ((A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) `) : ( ( ) ( ) Element of bool b1 : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ,X : ( ( ) ( ) set ) )) : ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( ) ( ) set ) -defined {{} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) ,1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of bool [:b1 : ( ( ) ( ) set ) ,{{} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) ,1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total ) Element of bool [:b1 : ( ( ) ( ) set ) ,NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = chi (A : ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ,X : ( ( ) ( ) set ) ) : ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( ) ( ) set ) -defined {{} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) ,1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of bool [:b1 : ( ( ) ( ) set ) ,{{} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) ,1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: HILBERT3:10
for a, b, x, y, x9, y9 being ( ( ) ( ) set ) st a : ( ( ) ( ) set ) <> b : ( ( ) ( ) set ) & (a : ( ( ) ( ) set ) ,b : ( ( ) ( ) set ) ) --> (x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like Function-like ) set ) = (a : ( ( ) ( ) set ) ,b : ( ( ) ( ) set ) ) --> (x9 : ( ( ) ( ) set ) ,y9 : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like Function-like ) set ) holds
( x : ( ( ) ( ) set ) = x9 : ( ( ) ( ) set ) & y : ( ( ) ( ) set ) = y9 : ( ( ) ( ) set ) ) ;

theorem :: HILBERT3:11
for a, b, x, y, X, Y being ( ( ) ( ) set ) st a : ( ( ) ( ) set ) <> b : ( ( ) ( ) set ) & x : ( ( ) ( ) set ) in X : ( ( ) ( ) set ) & y : ( ( ) ( ) set ) in Y : ( ( ) ( ) set ) holds
(a : ( ( ) ( ) set ) ,b : ( ( ) ( ) set ) ) --> (x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like Function-like ) set ) in product ((a : ( ( ) ( ) set ) ,b : ( ( ) ( ) set ) ) --> (X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) )) : ( ( ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) ;

theorem :: HILBERT3:12
for D being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like quasi_total ) ( non empty Relation-like 2 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of 2 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ,D : ( ( non empty ) ( non empty ) set ) ) ex d1, d2 being ( ( ) ( ) Element of D : ( ( non empty ) ( non empty ) set ) ) st f : ( ( Function-like quasi_total ) ( non empty Relation-like 2 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of 2 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) = (0 : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ) --> (d1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,d2 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like {0 : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) -defined NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of bool [:{0 : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: HILBERT3:13
for a, b, c, d being ( ( ) ( ) set ) st a : ( ( ) ( ) set ) <> b : ( ( ) ( ) set ) holds
((a : ( ( ) ( ) set ) ,b : ( ( ) ( ) set ) ) --> (c : ( ( ) ( ) set ) ,d : ( ( ) ( ) set ) )) : ( ( ) ( Relation-like Function-like ) set ) * ((a : ( ( ) ( ) set ) ,b : ( ( ) ( ) set ) ) --> (b : ( ( ) ( ) set ) ,a : ( ( ) ( ) set ) )) : ( ( ) ( Relation-like Function-like ) set ) : ( ( Relation-like ) ( Relation-like Function-like ) set ) = (a : ( ( ) ( ) set ) ,b : ( ( ) ( ) set ) ) --> (d : ( ( ) ( ) set ) ,c : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like Function-like ) set ) ;

theorem :: HILBERT3:14
for a, b, c, d being ( ( ) ( ) set )
for f being ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) st a : ( ( ) ( ) set ) <> b : ( ( ) ( ) set ) & c : ( ( ) ( ) set ) in dom f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( ) ( ) set ) & d : ( ( ) ( ) set ) in dom f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( ) ( ) set ) holds
f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) * ((a : ( ( ) ( ) set ) ,b : ( ( ) ( ) set ) ) --> (c : ( ( ) ( ) set ) ,d : ( ( ) ( ) set ) )) : ( ( ) ( Relation-like Function-like ) set ) : ( ( Relation-like ) ( Relation-like Function-like ) set ) = (a : ( ( ) ( ) set ) ,b : ( ( ) ( ) set ) ) --> ((f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . c : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,(f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . d : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like Function-like ) set ) ;

theorem :: HILBERT3:15
(0 : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ) --> (1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ,0 : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like {0 : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) -defined NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like one-to-one total quasi_total ) Element of bool [:{0 : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) } : ( ( ) ( non empty ) set ) ,NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) is ( ( Function-like quasi_total bijective ) ( non empty Relation-like 2 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) -defined 2 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of 2 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ) ;

begin

registration
let f, g be ( ( Relation-like Function-like one-to-one ) ( Relation-like Function-like one-to-one ) Function) ;
cluster [:f : ( ( Relation-like Function-like one-to-one ) ( Relation-like Function-like one-to-one ) set ) ,g : ( ( Relation-like Function-like one-to-one ) ( Relation-like Function-like one-to-one ) set ) :] : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) -> Relation-like Function-like one-to-one ;
end;

theorem :: HILBERT3:16
for A, B being ( ( non empty ) ( non empty ) set )
for C, D being ( ( ) ( ) set )
for f being ( ( Function-like quasi_total ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of C : ( ( ) ( ) set ) ,A : ( ( non empty ) ( non empty ) set ) )
for g being ( ( Function-like quasi_total ) ( Relation-like b4 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of D : ( ( ) ( ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) holds (pr1 (A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of bool [:[:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) * [:f : ( ( Function-like quasi_total ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of b3 : ( ( ) ( ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ,g : ( ( Function-like quasi_total ) ( Relation-like b4 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of b4 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) :] : ( ( Function-like quasi_total ) ( Relation-like [:b3 : ( ( ) ( ) set ) ,b4 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) -defined [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) -valued Function-like total quasi_total ) Element of bool [:[:b3 : ( ( ) ( ) set ) ,b4 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) ,[:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like [:b3 : ( ( ) ( ) set ) ,b4 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of bool [:[:b3 : ( ( ) ( ) set ) ,b4 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = f : ( ( Function-like quasi_total ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of b3 : ( ( ) ( ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) * (pr1 (C : ( ( ) ( ) set ) ,D : ( ( ) ( ) set ) )) : ( ( Function-like quasi_total ) ( Relation-like [:b3 : ( ( ) ( ) set ) ,b4 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) -defined b3 : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of bool [:[:b3 : ( ( ) ( ) set ) ,b4 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) ,b3 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like [:b3 : ( ( ) ( ) set ) ,b4 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of bool [:[:b3 : ( ( ) ( ) set ) ,b4 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: HILBERT3:17
for A, B being ( ( non empty ) ( non empty ) set )
for C, D being ( ( ) ( ) set )
for f being ( ( Function-like quasi_total ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of C : ( ( ) ( ) set ) ,A : ( ( non empty ) ( non empty ) set ) )
for g being ( ( Function-like quasi_total ) ( Relation-like b4 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of D : ( ( ) ( ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) holds (pr2 (A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of bool [:[:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) * [:f : ( ( Function-like quasi_total ) ( Relation-like b3 : ( ( ) ( ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of b3 : ( ( ) ( ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ,g : ( ( Function-like quasi_total ) ( Relation-like b4 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of b4 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) :] : ( ( Function-like quasi_total ) ( Relation-like [:b3 : ( ( ) ( ) set ) ,b4 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) -defined [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) -valued Function-like total quasi_total ) Element of bool [:[:b3 : ( ( ) ( ) set ) ,b4 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) ,[:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like [:b3 : ( ( ) ( ) set ) ,b4 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of bool [:[:b3 : ( ( ) ( ) set ) ,b4 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = g : ( ( Function-like quasi_total ) ( Relation-like b4 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of b4 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) * (pr2 (C : ( ( ) ( ) set ) ,D : ( ( ) ( ) set ) )) : ( ( Function-like quasi_total ) ( Relation-like [:b3 : ( ( ) ( ) set ) ,b4 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) -defined b4 : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of bool [:[:b3 : ( ( ) ( ) set ) ,b4 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) ,b4 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like [:b3 : ( ( ) ( ) set ) ,b4 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of bool [:[:b3 : ( ( ) ( ) set ) ,b4 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: HILBERT3:18
for g being ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) holds {} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) .. g : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) = {} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) ;

theorem :: HILBERT3:19
for f being ( ( Relation-like Function-like Function-yielding ) ( Relation-like Function-like Function-yielding V55() ) Function)
for g, h being ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) holds (f : ( ( Relation-like Function-like Function-yielding ) ( Relation-like Function-like Function-yielding V55() ) Function) .. g : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) * h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( Relation-like ) ( Relation-like Function-like ) set ) = (f : ( ( Relation-like Function-like Function-yielding ) ( Relation-like Function-like Function-yielding V55() ) Function) * h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) ) : ( ( Relation-like ) ( Relation-like Function-like Function-yielding V55() ) set ) .. (g : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) * h : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) ) : ( ( Relation-like ) ( Relation-like Function-like ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ;

theorem :: HILBERT3:20
for C being ( ( ) ( ) set )
for A being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like quasi_total ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined Funcs ({} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) ,b1 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) -valued Function-like quasi_total Function-yielding V55() ) Function of A : ( ( non empty ) ( non empty ) set ) , Funcs ({} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) ,C : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) )
for g being ( ( Function-like quasi_total ) ( ext-real empty trivial non proper V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding b2 : ( ( non empty ) ( non empty ) set ) -defined {} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) -valued Function-like one-to-one constant functional non total quasi_total Function-yielding V55() ) Function of A : ( ( non empty ) ( non empty ) set ) , {} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) ) holds rng (f : ( ( Function-like quasi_total ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined Funcs ({} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) ,b1 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) -valued Function-like quasi_total Function-yielding V55() ) Function of b2 : ( ( non empty ) ( non empty ) set ) , Funcs ({} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) ,b1 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) ) .. g : ( ( Function-like quasi_total ) ( ext-real empty trivial non proper V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding b2 : ( ( non empty ) ( non empty ) set ) -defined {} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) -valued Function-like one-to-one constant functional non total quasi_total Function-yielding V55() ) Function of b2 : ( ( non empty ) ( non empty ) set ) , {} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) = {{} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) } : ( ( ) ( non empty trivial functional ) set ) ;

theorem :: HILBERT3:21
for A, B, C being ( ( ) ( ) set ) st ( B : ( ( ) ( ) set ) = {} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) implies A : ( ( ) ( ) set ) = {} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) ) holds
for f being ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( ) ( ) set ) -defined Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) -valued Function-like quasi_total Function-yielding V55() ) Function of A : ( ( ) ( ) set ) , Funcs (B : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) )
for g being ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of A : ( ( ) ( ) set ) ,B : ( ( ) ( ) set ) ) holds rng (f : ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( ) ( ) set ) -defined Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) -valued Function-like quasi_total Function-yielding V55() ) Function of b1 : ( ( ) ( ) set ) , Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) ) .. g : ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) c= C : ( ( ) ( ) set ) ;

theorem :: HILBERT3:22
for A, B, C being ( ( ) ( ) set ) st ( not C : ( ( ) ( ) set ) = {} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) or B : ( ( ) ( ) set ) = {} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) or A : ( ( ) ( ) set ) = {} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) ) holds
for f being ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( ) ( ) set ) -defined Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) -valued Function-like quasi_total Function-yielding V55() ) Function of A : ( ( ) ( ) set ) , Funcs (B : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) ) holds dom (Frege f : ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( ) ( ) set ) -defined Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) -valued Function-like quasi_total Function-yielding V55() ) Function of b1 : ( ( ) ( ) set ) , Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) ) ) : ( ( Relation-like product (doms b4 : ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( ) ( ) set ) -defined Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) -valued Function-like quasi_total Function-yielding V55() ) Function of b1 : ( ( ) ( ) set ) , Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) -defined Function-like total Function-yielding ) ( Relation-like product (doms b4 : ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( ) ( ) set ) -defined Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) -valued Function-like quasi_total Function-yielding V55() ) Function of b1 : ( ( ) ( ) set ) , Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) -defined Function-like total Function-yielding V55() ) set ) : ( ( ) ( ) set ) = Funcs (A : ( ( ) ( ) set ) ,B : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) ;

theorem :: HILBERT3:23
for A, B, C being ( ( ) ( ) set ) st ( not C : ( ( ) ( ) set ) = {} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) or B : ( ( ) ( ) set ) = {} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) or A : ( ( ) ( ) set ) = {} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) ) holds
for f being ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( ) ( ) set ) -defined Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) -valued Function-like quasi_total Function-yielding V55() ) Function of A : ( ( ) ( ) set ) , Funcs (B : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) ) holds rng (Frege f : ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( ) ( ) set ) -defined Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) -valued Function-like quasi_total Function-yielding V55() ) Function of b1 : ( ( ) ( ) set ) , Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) ) ) : ( ( Relation-like product (doms b4 : ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( ) ( ) set ) -defined Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) -valued Function-like quasi_total Function-yielding V55() ) Function of b1 : ( ( ) ( ) set ) , Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) -defined Function-like total Function-yielding ) ( Relation-like product (doms b4 : ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( ) ( ) set ) -defined Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) -valued Function-like quasi_total Function-yielding V55() ) Function of b1 : ( ( ) ( ) set ) , Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) -defined Function-like total Function-yielding V55() ) set ) : ( ( ) ( ) set ) c= Funcs (A : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) ;

theorem :: HILBERT3:24
for A, B, C being ( ( ) ( ) set ) st ( not C : ( ( ) ( ) set ) = {} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) or B : ( ( ) ( ) set ) = {} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) or A : ( ( ) ( ) set ) = {} : ( ( ) ( ext-real empty trivial V7() V8() V9() V11() V12() V13() complex V15() integer Relation-like non-empty empty-yielding Function-like one-to-one constant functional Function-yielding V55() ) set ) ) holds
for f being ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( ) ( ) set ) -defined Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) -valued Function-like quasi_total Function-yielding V55() ) Function of A : ( ( ) ( ) set ) , Funcs (B : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) ) holds Frege f : ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( ) ( ) set ) -defined Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) -valued Function-like quasi_total Function-yielding V55() ) Function of b1 : ( ( ) ( ) set ) , Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) ) : ( ( Relation-like product (doms b4 : ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( ) ( ) set ) -defined Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) -valued Function-like quasi_total Function-yielding V55() ) Function of b1 : ( ( ) ( ) set ) , Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) -defined Function-like total Function-yielding ) ( Relation-like product (doms b4 : ( ( Function-like quasi_total ) ( Relation-like b1 : ( ( ) ( ) set ) -defined Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) -valued Function-like quasi_total Function-yielding V55() ) Function of b1 : ( ( ) ( ) set ) , Funcs (b2 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) -defined Function-like total Function-yielding V55() ) set ) is ( ( Function-like quasi_total ) ( Relation-like Funcs (b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) -defined Funcs (b1 : ( ( ) ( ) set ) ,b3 : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) -valued Function-like quasi_total Function-yielding V55() ) Function of Funcs (A : ( ( ) ( ) set ) ,B : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) , Funcs (A : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) ) ;

begin

registration
let A, B be ( ( ) ( ) set ) ;
let P be ( ( Function-like quasi_total bijective ) ( Relation-like A : ( ( ) ( ) set ) -defined A : ( ( ) ( ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of A : ( ( ) ( ) set ) ) ;
let Q be ( ( Function-like quasi_total bijective ) ( Relation-like B : ( ( ) ( ) set ) -defined B : ( ( ) ( ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of B : ( ( ) ( ) set ) ) ;
cluster [:P : ( ( Function-like quasi_total bijective ) ( Relation-like A : ( ( ) ( ) set ) -defined A : ( ( ) ( ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Element of bool [:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ,Q : ( ( Function-like quasi_total bijective ) ( Relation-like B : ( ( ) ( ) set ) -defined B : ( ( ) ( ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Element of bool [:B : ( ( ) ( ) set ) ,B : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( Relation-like Function-like ) ( Relation-like Function-like one-to-one ) set ) -> Function-like quasi_total bijective for ( ( Function-like quasi_total ) ( Relation-like [:A : ( ( ) ( ) set ) ,B : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) -defined [:A : ( ( ) ( ) set ) ,B : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) -valued Function-like total quasi_total ) Function of [:A : ( ( ) ( ) set ) ,B : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) ,[:A : ( ( ) ( ) set ) ,B : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) ) ;
end;

theorem :: HILBERT3:25
for A, B being ( ( ) ( ) set )
for P being ( ( Function-like quasi_total bijective ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b1 : ( ( ) ( ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of A : ( ( ) ( ) set ) )
for Q being ( ( Function-like quasi_total bijective ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b2 : ( ( ) ( ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of B : ( ( ) ( ) set ) ) holds [:P : ( ( Function-like quasi_total bijective ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b1 : ( ( ) ( ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of b1 : ( ( ) ( ) set ) ) ,Q : ( ( Function-like quasi_total bijective ) ( Relation-like b2 : ( ( ) ( ) set ) -defined b2 : ( ( ) ( ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of b2 : ( ( ) ( ) set ) ) :] : ( ( Function-like quasi_total ) ( Relation-like [:b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) -defined [:b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Element of bool [:[:b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) ,[:b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) is bijective ;

definition
let A, B be ( ( non empty ) ( non empty ) set ) ;
let P be ( ( Function-like quasi_total bijective ) ( non empty Relation-like A : ( ( non empty ) ( non empty ) set ) -defined A : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of A : ( ( non empty ) ( non empty ) set ) ) ;
let Q be ( ( Function-like quasi_total ) ( non empty Relation-like B : ( ( non empty ) ( non empty ) set ) -defined B : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of B : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) ;
func P => Q -> ( ( Function-like quasi_total ) ( non empty Relation-like Funcs (A : ( ( ) ( ) set ) ,B : ( ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of A : ( ( ) ( ) set ) ,B : ( ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) set ) ) -defined Funcs (A : ( ( ) ( ) set ) ,B : ( ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of A : ( ( ) ( ) set ) ,B : ( ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) set ) ) -valued Function-like total quasi_total Function-yielding V55() ) Function of Funcs (A : ( ( ) ( ) set ) ,B : ( ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of A : ( ( ) ( ) set ) ,B : ( ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) set ) ) , Funcs (A : ( ( ) ( ) set ) ,B : ( ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of A : ( ( ) ( ) set ) ,B : ( ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) set ) ) ) means :: HILBERT3:def 1
 for f being ( ( Function-like quasi_total ) ( non empty Relation-like A : ( ( ) ( ) set ) -defined B : ( ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) set ) -valued Function-like total quasi_total ) Function of A : ( ( ) ( ) set ) ,B : ( ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) set ) ) holds it : ( ( ) ( Relation-like the U1 of A : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined Function-like total Function-yielding V55() ) ManySortedFunction of the U11 of B : ( ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) set ) : ( ( Relation-like the U1 of A : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like the U1 of A : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined Function-like total ) set ) , the U11 of P : ( ( Function-like quasi_total bijective ) ( Relation-like A : ( ( ) ( ) set ) -defined A : ( ( ) ( ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Element of bool [:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Relation-like the U1 of A : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like the U1 of A : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined Function-like total ) set ) ) . f : ( ( Function-like quasi_total ) ( non empty Relation-like A : ( ( non empty ) ( non empty ) set ) -defined B : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) set ) = (Q : ( ( Function-like quasi_total bijective ) ( Relation-like B : ( ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) set ) -defined B : ( ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Element of bool [:B : ( ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) set ) ,B : ( ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty Relation-like A : ( ( non empty ) ( non empty ) set ) -defined B : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like A : ( ( ) ( ) set ) -defined A : ( ( non empty ) ( non empty ) set ) -defined B : ( ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) set ) -valued Function-like ) Element of bool [:A : ( ( ) ( ) set ) ,B : ( ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) * (P : ( ( Function-like quasi_total bijective ) ( Relation-like A : ( ( ) ( ) set ) -defined A : ( ( ) ( ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Element of bool [:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ") : ( ( Function-like quasi_total bijective ) ( Relation-like A : ( ( ) ( ) set ) -defined A : ( ( ) ( ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Element of bool [:A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like A : ( ( ) ( ) set ) -defined B : ( ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) set ) -valued Function-like ) Element of bool [:A : ( ( ) ( ) set ) ,B : ( ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like A : ( ( ) ( ) set ) -defined Function-like total ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

registration
let A, B be ( ( non empty ) ( non empty ) set ) ;
let P be ( ( Function-like quasi_total bijective ) ( non empty Relation-like A : ( ( non empty ) ( non empty ) set ) -defined A : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of A : ( ( non empty ) ( non empty ) set ) ) ;
let Q be ( ( Function-like quasi_total bijective ) ( non empty Relation-like B : ( ( non empty ) ( non empty ) set ) -defined B : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of B : ( ( non empty ) ( non empty ) set ) ) ;
cluster P : ( ( Function-like quasi_total bijective ) ( non empty Relation-like A : ( ( non empty ) ( non empty ) set ) -defined A : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Element of bool [:A : ( ( non empty ) ( non empty ) set ) ,A : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) => Q : ( ( Function-like quasi_total bijective ) ( non empty Relation-like B : ( ( non empty ) ( non empty ) set ) -defined B : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Element of bool [:B : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like Funcs (A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) -defined Funcs (A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V55() ) Function of Funcs (A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) , Funcs (A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) ) -> Function-like quasi_total bijective ;
end;

theorem :: HILBERT3:26
for A, B being ( ( non empty ) ( non empty ) set )
for P being ( ( Function-like quasi_total bijective ) ( non empty Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of A : ( ( non empty ) ( non empty ) set ) )
for Q being ( ( Function-like quasi_total bijective ) ( non empty Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of B : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like quasi_total ) ( non empty Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) ) holds ((P : ( ( Function-like quasi_total bijective ) ( non empty Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of b1 : ( ( non empty ) ( non empty ) set ) ) => Q : ( ( Function-like quasi_total bijective ) ( non empty Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like one-to-one total quasi_total onto bijective Function-yielding V55() ) Function of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ") : ( ( Function-like quasi_total bijective ) ( non empty Relation-like Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like one-to-one total quasi_total onto bijective Function-yielding V55() ) Element of bool [:(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) . f : ( ( Function-like quasi_total ) ( non empty Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) set ) = ((Q : ( ( Function-like quasi_total bijective ) ( non empty Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of b2 : ( ( non empty ) ( non empty ) set ) ) ") : ( ( Function-like quasi_total bijective ) ( non empty Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Element of bool [:b2 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) * f : ( ( Function-like quasi_total ) ( non empty Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) * P : ( ( Function-like quasi_total bijective ) ( non empty Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like ) ( non empty Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: HILBERT3:27
for A, B being ( ( non empty ) ( non empty ) set )
for P being ( ( Function-like quasi_total bijective ) ( non empty Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of A : ( ( non empty ) ( non empty ) set ) )
for Q being ( ( Function-like quasi_total bijective ) ( non empty Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of B : ( ( non empty ) ( non empty ) set ) ) holds (P : ( ( Function-like quasi_total bijective ) ( non empty Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of b1 : ( ( non empty ) ( non empty ) set ) ) => Q : ( ( Function-like quasi_total bijective ) ( non empty Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like one-to-one total quasi_total onto bijective Function-yielding V55() ) Function of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) " : ( ( Function-like quasi_total bijective ) ( non empty Relation-like Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like one-to-one total quasi_total onto bijective Function-yielding V55() ) Element of bool [:(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = (P : ( ( Function-like quasi_total bijective ) ( non empty Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of b1 : ( ( non empty ) ( non empty ) set ) ) ") : ( ( Function-like quasi_total bijective ) ( non empty Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) => (Q : ( ( Function-like quasi_total bijective ) ( non empty Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of b2 : ( ( non empty ) ( non empty ) set ) ) ") : ( ( Function-like quasi_total bijective ) ( non empty Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Element of bool [:b2 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like one-to-one total quasi_total onto bijective Function-yielding V55() ) Function of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) , Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: HILBERT3:28
for A, B, C being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like quasi_total ) ( non empty Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Funcs (b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V55() ) Function of A : ( ( non empty ) ( non empty ) set ) , Funcs (B : ( ( non empty ) ( non empty ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) )
for g being ( ( Function-like quasi_total ) ( non empty Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of A : ( ( non empty ) ( non empty ) set ) ,B : ( ( non empty ) ( non empty ) set ) )
for P being ( ( Function-like quasi_total bijective ) ( non empty Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of B : ( ( non empty ) ( non empty ) set ) )
for Q being ( ( Function-like quasi_total bijective ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of C : ( ( non empty ) ( non empty ) set ) ) holds ((P : ( ( Function-like quasi_total bijective ) ( non empty Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of b2 : ( ( non empty ) ( non empty ) set ) ) => Q : ( ( Function-like quasi_total bijective ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of b3 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like Funcs (b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) -defined Funcs (b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like one-to-one total quasi_total onto bijective Function-yielding V55() ) Function of Funcs (b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) , Funcs (b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) * f : ( ( Function-like quasi_total ) ( non empty Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Funcs (b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V55() ) Function of b1 : ( ( non empty ) ( non empty ) set ) , Funcs (b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( non empty Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Funcs (b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V55() ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,(Funcs (b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) .. (P : ( ( Function-like quasi_total bijective ) ( non empty Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of b2 : ( ( non empty ) ( non empty ) set ) ) * g : ( ( Function-like quasi_total ) ( non empty Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of bool [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) = Q : ( ( Function-like quasi_total bijective ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of b3 : ( ( non empty ) ( non empty ) set ) ) * (f : ( ( Function-like quasi_total ) ( non empty Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined Funcs (b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V55() ) Function of b1 : ( ( non empty ) ( non empty ) set ) , Funcs (b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) .. g : ( ( Function-like quasi_total ) ( non empty Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( Relation-like ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) set ) ;

begin

definition
mode SetValuation is ( ( Relation-like V19() NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like V19() non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ManySortedSet of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ;
end;

definition
let V be ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ;
func SetVal V -> ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) means :: HILBERT3:def 2
( it : ( ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) set ) . VERUM : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( ) set ) = 1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) & ( for n being ( ( ) ( ext-real V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) holds it : ( ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) set ) . (prop n : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( ) set ) = V : ( ( ) ( ) set ) . n : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( ) set ) ) & ( for p, q being ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) holds
( it : ( ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) set ) . (p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) '&' q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( ) set ) = [:(it : ( ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) set ) . p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( ) set ) ,(it : ( ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) set ) . q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) & it : ( ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) set ) . (p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( ) set ) = Funcs ((it : ( ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) set ) . p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( ) set ) ,(it : ( ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) set ) . q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional ) set ) ) ) );
end;

definition
let V be ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ;
let p be ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ;
func SetVal (V,p) -> ( ( ) ( ) set ) equals :: HILBERT3:def 3
(SetVal V : ( ( ) ( ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . p : ( ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) set ) : ( ( ) ( ) set ) ;
end;

registration
let V be ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ;
let p be ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ;
cluster SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( ) set ) -> non empty ;
end;

theorem :: HILBERT3:29
for V being ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) holds SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,VERUM : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) = 1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: HILBERT3:30
for n being ( ( ) ( ext-real V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) )
for V being ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) holds SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(prop n : ( ( ) ( ext-real V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) = V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) . n : ( ( ) ( ext-real V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ;

theorem :: HILBERT3:31
for p, q being ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) )
for V being ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) holds SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) '&' q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) = [:(SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) ;

theorem :: HILBERT3:32
for p, q being ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) )
for V being ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) holds SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) = Funcs ((SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) ;

registration
let V be ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ;
let p, q be ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ;
cluster SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,(p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -> functional ;
end;

registration
let V be ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ;
let p, q, r be ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ;
cluster -> Function-yielding for ( ( ) ( ) Element of SetVal (V : ( ( ) ( ) set ) ,(p : ( ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) set ) => (q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => r : ( ( Function-like quasi_total bijective ) ( non empty Relation-like p : ( ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) set ) -defined p : ( ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Element of bool [:p : ( ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) set ) ,p : ( ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) ( Relation-like V : ( ( ) ( ) set ) -defined Function-like total ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( ) set ) ) ;
end;

registration
let V be ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ;
let p, q, r be ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ;
cluster non empty Relation-like SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,(p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) -defined SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,(p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => r : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total Function-yielding V55() for ( ( ) ( ) Element of bool [:(SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,(p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty functional ) set ) ,(SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,(p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => r : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;
cluster Relation-like Function-like Function-yielding V55() for ( ( ) ( ) Element of SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,(p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => (q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => r : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) ) ;
end;

begin

definition
let V be ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ;
mode Permutation of V -> ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) means :: HILBERT3:def 4
( dom it : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( ) set ) = NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) & ( for n being ( ( ) ( ext-real V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) holds it : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . n : ( ( ) ( ext-real V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) is ( ( Function-like quasi_total bijective ) ( non empty Relation-like V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) . b1 : ( ( ) ( ext-real V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) -defined V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) . b1 : ( ( ) ( ext-real V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) . n : ( ( ) ( ext-real V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) );
end;

definition
let V be ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ;
let P be ( ( ) ( Relation-like Function-like ) Permutation of V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ;
func Perm P -> ( ( ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total Function-yielding V55() ) ManySortedFunction of SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) , SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) means :: HILBERT3:def 5
( it : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . VERUM : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( Function-like quasi_total ) ( Relation-like (SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . VERUM : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( ) set ) -defined (SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . VERUM : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) Element of bool [:((SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . VERUM : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( ) set ) ,((SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . VERUM : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = id 1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( total ) ( non empty Relation-like 1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) -defined 1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) -valued Function-like one-to-one total quasi_total onto bijective V35() V37() V38() V42() ) Element of bool [:1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( ext-real non empty V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) & ( for n being ( ( ) ( ext-real V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) holds it : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . (prop n : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( Function-like quasi_total ) ( Relation-like (SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . (prop b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( ) set ) -defined (SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . (prop b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) Element of bool [:((SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . (prop b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( ) set ) ,((SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . (prop b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = P : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . n : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( ) set ) ) & ( for p, q being ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ex p9 being ( ( Function-like quasi_total bijective ) ( non empty Relation-like SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) ex q9 being ( ( Function-like quasi_total bijective ) ( non empty Relation-like SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) st
( p9 : ( ( Function-like quasi_total bijective ) ( non empty Relation-like SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) = it : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( Function-like quasi_total ) ( Relation-like (SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( ) set ) -defined (SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) Element of bool [:((SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( ) set ) ,((SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) & q9 : ( ( Function-like quasi_total bijective ) ( non empty Relation-like SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) = it : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( Function-like quasi_total ) ( Relation-like (SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( ) set ) -defined (SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) Element of bool [:((SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( ) set ) ,((SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) & it : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . (p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) '&' q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( Function-like quasi_total ) ( Relation-like (SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . (b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) '&' b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( ) set ) -defined (SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . (b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) '&' b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) Element of bool [:((SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . (b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) '&' b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( ) set ) ,((SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . (b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) '&' b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = [:p9 : ( ( Function-like quasi_total bijective ) ( non empty Relation-like SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) ,q9 : ( ( Function-like quasi_total bijective ) ( non empty Relation-like SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) :] : ( ( Function-like quasi_total ) ( non empty Relation-like [:(SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) -defined [:(SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) -valued Function-like one-to-one total quasi_total ) Element of bool [:[:(SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) ,[:(SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) & it : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . (p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( Function-like quasi_total ) ( Relation-like (SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . (b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( ) set ) -defined (SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . (b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) Element of bool [:((SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . (b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( ) set ) ,((SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . (b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = p9 : ( ( Function-like quasi_total bijective ) ( non empty Relation-like SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) => q9 : ( ( Function-like quasi_total bijective ) ( non empty Relation-like SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like Funcs ((SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) -defined Funcs ((SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) -valued Function-like total quasi_total Function-yielding V55() ) Function of Funcs ((SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) , Funcs ((SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) ) ) );
end;

definition
let V be ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ;
let P be ( ( ) ( Relation-like Function-like ) Permutation of V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ;
let p be ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ;
func Perm (P,p) -> ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) equals :: HILBERT3:def 6
(Perm P : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total Function-yielding V55() ) ManySortedFunction of SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) , SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) . p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( Function-like quasi_total ) ( Relation-like (SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( ) set ) -defined (SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) Element of bool [:((SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( ) set ) ,((SetVal V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ( non empty Relation-like HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) -defined Function-like total ) ManySortedSet of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) . p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

theorem :: HILBERT3:33
for V being ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation)
for P being ( ( ) ( Relation-like Function-like ) Permutation of V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) holds Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of b1 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ,VERUM : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b1 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,VERUM : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b1 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,VERUM : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of SetVal (b1 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,VERUM : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b1 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,VERUM : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) = id (SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,VERUM : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) : ( ( total ) ( non empty Relation-like SetVal (b1 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,VERUM : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b1 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,VERUM : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective V35() V37() V38() V42() ) Element of bool [:(SetVal (b1 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,VERUM : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (b1 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,VERUM : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: HILBERT3:34
for n being ( ( ) ( ext-real V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) )
for V being ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation)
for P being ( ( ) ( Relation-like Function-like ) Permutation of V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) holds Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of b2 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ,(prop n : ( ( ) ( ext-real V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b2 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(prop b1 : ( ( ) ( ext-real V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b2 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(prop b1 : ( ( ) ( ext-real V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of SetVal (b2 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(prop b1 : ( ( ) ( ext-real V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b2 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(prop b1 : ( ( ) ( ext-real V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) = P : ( ( ) ( Relation-like Function-like ) Permutation of b2 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) . n : ( ( ) ( ext-real V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) ;

theorem :: HILBERT3:35
for p, q being ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) )
for V being ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation)
for P being ( ( ) ( Relation-like Function-like ) Permutation of V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) holds Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ,(p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) '&' q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) '&' b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) '&' b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) '&' b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) '&' b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) = [:(Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ,p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) ,(Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ,q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) :] : ( ( Function-like quasi_total ) ( non empty Relation-like [:(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) -defined [:(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) -valued Function-like total quasi_total ) Element of bool [:[:(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) ,[:(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: HILBERT3:36
for p, q being ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) )
for V being ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation)
for P being ( ( ) ( Relation-like Function-like ) Permutation of V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) )
for p9 being ( ( Function-like quasi_total bijective ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) )
for q9 being ( ( Function-like quasi_total bijective ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) st p9 : ( ( Function-like quasi_total bijective ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) = Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ,p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) & q9 : ( ( Function-like quasi_total bijective ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) = Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ,q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) holds
Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ,(p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total quasi_total Function-yielding V55() ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) ) = p9 : ( ( Function-like quasi_total bijective ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) => q9 : ( ( Function-like quasi_total bijective ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Permutation of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like Funcs ((SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) -defined Funcs ((SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) -valued Function-like one-to-one total quasi_total onto bijective Function-yielding V55() ) Function of Funcs ((SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) , Funcs ((SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty functional ) FUNCTION_DOMAIN of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) ;

registration
let V be ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ;
let P be ( ( ) ( Relation-like Function-like ) Permutation of V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ;
let p be ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ;
cluster Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) ,p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ,p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) -> Function-like quasi_total bijective ;
end;

theorem :: HILBERT3:37
for p, q being ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) )
for V being ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation)
for P being ( ( ) ( Relation-like Function-like ) Permutation of V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) )
for g being ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) holds (Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ,(p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like one-to-one total quasi_total onto bijective Function-yielding V55() ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) ) . g : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) set ) = ((Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ,q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) * g : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of bool [:(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) * ((Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ,p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) ") : ( ( Function-like quasi_total bijective ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Element of bool [:(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of bool [:(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: HILBERT3:38
for p, q being ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) )
for V being ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation)
for P being ( ( ) ( Relation-like Function-like ) Permutation of V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) )
for g being ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) holds ((Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ,(p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like one-to-one total quasi_total onto bijective Function-yielding V55() ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) ) ") : ( ( Function-like quasi_total bijective ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like one-to-one total quasi_total onto bijective Function-yielding V55() ) Element of bool [:(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty functional ) set ) ,(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) . g : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) set ) = (((Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ,q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) ") : ( ( Function-like quasi_total bijective ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Element of bool [:(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) * g : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of bool [:(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) * (Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ,p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of bool [:(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: HILBERT3:39
for p, q being ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) )
for V being ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation)
for P being ( ( ) ( Relation-like Function-like ) Permutation of V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) )
for f, g being ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) st f : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) = (Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ,(p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like one-to-one total quasi_total onto bijective Function-yielding V55() ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) ) . g : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) set ) holds
(Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ,q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) * g : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of bool [:(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = f : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) * (Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ,p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of bool [:(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) ,(SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) )) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: HILBERT3:40
for p, q being ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) )
for V being ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation)
for P being ( ( ) ( Relation-like Function-like ) Permutation of V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) )
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) is_a_fixpoint_of Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ,p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) holds
for f being ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) st f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) is_a_fixpoint_of Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ,(p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like one-to-one total quasi_total onto bijective Function-yielding V55() ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,(b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty functional ) set ) ) holds
f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . x : ( ( ) ( ) set ) : ( ( ) ( ) set ) is_a_fixpoint_of Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ,q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) ;

begin

definition
let p be ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ;
attr p is canonical means :: HILBERT3:def 7
 for V being ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ex x being ( ( ) ( ) set ) st
for P being ( ( ) ( Relation-like Function-like ) Permutation of V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) holds x : ( ( ) ( ) set ) is_a_fixpoint_of Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of b1 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ,p : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b1 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,p : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( ) ( ) set ) -defined SetVal (b1 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,p : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) Function of SetVal (b1 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,p : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( ) ( ) set ) , SetVal (b1 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,p : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( ) ( ) set ) ) ;
end;

registration
cluster VERUM : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) -> canonical ;
end;

registration
let p, q be ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ;
cluster p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => (q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) -> canonical ;
cluster (p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) '&' q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) -> canonical ;
cluster (p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) '&' q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) -> canonical ;
cluster p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => (q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => (p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) '&' q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) -> canonical ;
end;

registration
let p, q, r be ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ;
cluster (p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => (q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => r : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => ((p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => (p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => r : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) -> canonical ;
end;

theorem :: HILBERT3:41
for p, q being ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) st p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) is canonical & p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) is canonical holds
q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) is canonical ;

theorem :: HILBERT3:42
for p being ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) st p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) in HP_TAUT : ( ( ) ( functional Hilbert_theory ) Element of bool HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) : ( ( ) ( non empty ) set ) ) holds
p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) is canonical ;

registration
cluster Relation-like Function-like V51() canonical for ( ( ) ( ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ;
end;

begin

definition
let p be ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ;
attr p is pseudo-canonical means :: HILBERT3:def 8
 for V being ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation)
for P being ( ( ) ( Relation-like Function-like ) Permutation of V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ex x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) is_a_fixpoint_of Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of b1 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ,p : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b1 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,p : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( ) ( ) set ) -defined SetVal (b1 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,p : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( ) ( ) set ) -valued Function-like total quasi_total ) Function of SetVal (b1 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,p : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( ) ( ) set ) , SetVal (b1 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,p : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) set ) ) : ( ( ) ( ) set ) ) ;
end;

registration
cluster canonical -> pseudo-canonical for ( ( ) ( ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ;
end;

theorem :: HILBERT3:43
for p, q being ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) st p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) is pseudo-canonical & p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) is pseudo-canonical holds
q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) is pseudo-canonical ;

theorem :: HILBERT3:44
for p, q being ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) )
for V being ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation)
for P being ( ( ) ( Relation-like Function-like ) Permutation of V : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) st ex f being ( ( ) ( ) set ) st f : ( ( ) ( ) set ) is_a_fixpoint_of Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ,p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b1 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) & ( for f being ( ( ) ( ) set ) holds not f : ( ( ) ( ) set ) is_a_fixpoint_of Perm (P : ( ( ) ( Relation-like Function-like ) Permutation of b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ) ,q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -defined SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total quasi_total onto bijective ) Function of SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) , SetVal (b3 : ( ( Relation-like non-empty NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( non empty Relation-like non-empty non empty-yielding NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) SetValuation) ,b2 : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( non empty ) set ) ) ) holds
not p : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => q : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) is pseudo-canonical ;

theorem :: HILBERT3:45
for a, b being ( ( ) ( ext-real V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) st a : ( ( ) ( ext-real V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) <> b : ( ( ) ( ext-real V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) holds
not (((prop a : ( ( ) ( ext-real V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => (prop b : ( ( ) ( ext-real V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => (prop a : ( ( ) ( ext-real V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) => (prop a : ( ( ) ( ext-real V7() V8() V9() V13() complex V15() integer ) Element of NAT : ( ( ) ( non empty V7() V8() V9() ) Element of bool REAL : ( ( ) ( non empty V43() ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) : ( ( ) ( Relation-like Function-like V51() ) Element of HP-WFF : ( ( ) ( non empty functional with_VERUM with_implication with_conjunction with_propositional_variables HP-closed ) set ) ) is pseudo-canonical ;