:: INTPRO_1 semantic presentation

begin

definition
let E be ( ( ) ( ) set ) ;
attr E is with_FALSUM means :: INTPRO_1:def 1
<*0 : ( ( ) ( V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) *> : ( ( ) ( non empty V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V19( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() V39(1 : ( ( ) ( non empty V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like ) FinSequence of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) in E : ( ( V15() ) ( V15() ) set ) ;
end;

definition
let E be ( ( ) ( ) set ) ;
attr E is with_int_implication means :: INTPRO_1:def 2
 for p, q being ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) FinSequence) st p : ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) FinSequence) in E : ( ( V15() ) ( V15() ) set ) & q : ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) FinSequence) in E : ( ( V15() ) ( V15() ) set ) holds
(<*1 : ( ( ) ( non empty V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) *> : ( ( ) ( non empty V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V19( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() V39(1 : ( ( ) ( non empty V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like ) FinSequence of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ^ p : ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) FinSequence) ) : ( ( V15() Function-like FinSequence-like ) ( non empty V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) set ) ^ q : ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) FinSequence) : ( ( V15() Function-like FinSequence-like ) ( non empty V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) set ) in E : ( ( V15() ) ( V15() ) set ) ;
end;

definition
let E be ( ( ) ( ) set ) ;
attr E is with_int_conjunction means :: INTPRO_1:def 3
 for p, q being ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) FinSequence) st p : ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) FinSequence) in E : ( ( V15() ) ( V15() ) set ) & q : ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) FinSequence) in E : ( ( V15() ) ( V15() ) set ) holds
(<*2 : ( ( ) ( non empty V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) *> : ( ( ) ( non empty V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V19( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() V39(1 : ( ( ) ( non empty V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like ) FinSequence of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ^ p : ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) FinSequence) ) : ( ( V15() Function-like FinSequence-like ) ( non empty V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) set ) ^ q : ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) FinSequence) : ( ( V15() Function-like FinSequence-like ) ( non empty V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) set ) in E : ( ( V15() ) ( V15() ) set ) ;
end;

definition
let E be ( ( ) ( ) set ) ;
attr E is with_int_disjunction means :: INTPRO_1:def 4
 for p, q being ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) FinSequence) st p : ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) FinSequence) in E : ( ( V15() ) ( V15() ) set ) & q : ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) FinSequence) in E : ( ( V15() ) ( V15() ) set ) holds
(<*3 : ( ( ) ( non empty V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) *> : ( ( ) ( non empty V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V19( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() V39(1 : ( ( ) ( non empty V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like ) FinSequence of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ^ p : ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) FinSequence) ) : ( ( V15() Function-like FinSequence-like ) ( non empty V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) set ) ^ q : ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) FinSequence) : ( ( V15() Function-like FinSequence-like ) ( non empty V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) set ) in E : ( ( V15() ) ( V15() ) set ) ;
end;

definition
let E be ( ( ) ( ) set ) ;
attr E is with_int_propositional_variables means :: INTPRO_1:def 5
 for n being ( ( ) ( V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) holds <*(5 : ( ( ) ( non empty V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) + (2 : ( ( ) ( non empty V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) * n : ( ( ) ( V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) *> : ( ( ) ( non empty V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V19( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() V39(1 : ( ( ) ( non empty V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like ) FinSequence of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) in E : ( ( V15() ) ( V15() ) set ) ;
end;

definition
let E be ( ( ) ( ) set ) ;
attr E is with_modal_operator means :: INTPRO_1:def 6
 for p being ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) FinSequence) st p : ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) FinSequence) in E : ( ( V15() ) ( V15() ) set ) holds
<*6 : ( ( ) ( non empty V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) *> : ( ( ) ( non empty V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V19( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() V39(1 : ( ( ) ( non empty V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like ) FinSequence of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ^ p : ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) FinSequence) : ( ( V15() Function-like FinSequence-like ) ( non empty V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) set ) in E : ( ( V15() ) ( V15() ) set ) ;
end;

definition
let E be ( ( ) ( ) set ) ;
attr E is MC-closed means :: INTPRO_1:def 7
( E : ( ( V15() ) ( V15() ) set ) c= NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) * : ( ( ) ( non empty functional FinSequence-membered ) set ) & E : ( ( V15() ) ( V15() ) set ) is with_FALSUM & E : ( ( V15() ) ( V15() ) set ) is with_int_implication & E : ( ( V15() ) ( V15() ) set ) is with_int_conjunction & E : ( ( V15() ) ( V15() ) set ) is with_int_disjunction & E : ( ( V15() ) ( V15() ) set ) is with_int_propositional_variables & E : ( ( V15() ) ( V15() ) set ) is with_modal_operator );
end;

registration
cluster MC-closed -> non empty with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator for ( ( ) ( ) set ) ;
cluster with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator -> MC-closed for ( ( ) ( ) Element of K6((NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) *) : ( ( ) ( non empty functional FinSequence-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
func MC-wff -> ( ( ) ( ) set ) means :: INTPRO_1:def 8
( it : ( ( ) ( ) set ) is MC-closed & ( for E being ( ( ) ( ) set ) st E : ( ( ) ( ) set ) is MC-closed holds
it : ( ( ) ( ) set ) c= E : ( ( ) ( ) set ) ) );
end;

registration
cluster MC-wff : ( ( ) ( ) set ) -> MC-closed ;
end;

registration
cluster non empty MC-closed for ( ( ) ( ) set ) ;
end;

registration
cluster MC-wff : ( ( ) ( non empty with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) -> functional ;
end;

registration
cluster -> FinSequence-like for ( ( ) ( ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ;
end;

definition
mode MC-formula is ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ;
end;

definition
func FALSUM -> ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) equals :: INTPRO_1:def 9
<*0 : ( ( ) ( V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) *> : ( ( ) ( non empty V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V19( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() V39(1 : ( ( ) ( non empty V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like ) FinSequence of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;
let p, q be ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ;
func p => q -> ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) equals :: INTPRO_1:def 10
(<*1 : ( ( ) ( non empty V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) *> : ( ( ) ( non empty V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V19( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() V39(1 : ( ( ) ( non empty V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like ) FinSequence of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ^ p : ( ( ) ( ) set ) ) : ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) set ) ^ q : ( ( V15() Function-like ) ( V15() Function-like ) set ) : ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) set ) ;
func p '&' q -> ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) equals :: INTPRO_1:def 11
(<*2 : ( ( ) ( non empty V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) *> : ( ( ) ( non empty V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V19( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() V39(1 : ( ( ) ( non empty V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like ) FinSequence of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ^ p : ( ( ) ( ) set ) ) : ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) set ) ^ q : ( ( V15() Function-like ) ( V15() Function-like ) set ) : ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) set ) ;
func p 'or' q -> ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) equals :: INTPRO_1:def 12
(<*3 : ( ( ) ( non empty V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) *> : ( ( ) ( non empty V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V19( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() V39(1 : ( ( ) ( non empty V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like ) FinSequence of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ^ p : ( ( ) ( ) set ) ) : ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) set ) ^ q : ( ( V15() Function-like ) ( V15() Function-like ) set ) : ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) set ) ;
end;

definition
let p be ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ;
func Nes p -> ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) equals :: INTPRO_1:def 13
<*6 : ( ( ) ( non empty V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) *> : ( ( ) ( non empty V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) V19( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() V39(1 : ( ( ) ( non empty V4() V5() V6() V10() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like ) FinSequence of NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) ^ p : ( ( ) ( ) set ) : ( ( V15() Function-like FinSequence-like ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) set ) ;
end;

definition
let T be ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;
attr T is IPC_theory means :: INTPRO_1:def 14
 for p, q, r being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds
( p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & FALSUM : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & ( p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in T : ( ( ) ( ) set ) & p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) implies q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in T : ( ( ) ( ) set ) ) );
end;

definition
let X be ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;
func CnIPC X -> ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) means :: INTPRO_1:def 15
 for r being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds
( r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in it : ( ( V15() Function-like ) ( V15() Function-like ) set ) iff for T being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) st T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) is IPC_theory & X : ( ( ) ( ) set ) c= T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds
r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) );
end;

definition
func IPC-Taut -> ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) equals :: INTPRO_1:def 16
 CnIPC ({} MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( empty proper V4() V5() V6() V8() V9() V10() Function-like functional FinSequence-membered ) Element of K6(MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;
end;

definition
let p be ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ;
func neg p -> ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) equals :: INTPRO_1:def 17
p : ( ( ) ( ) set ) => FALSUM : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ;
end;

definition
func IVERUM -> ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) equals :: INTPRO_1:def 18
FALSUM : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => FALSUM : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ;
end;

theorem :: INTPRO_1:1
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) )
for p, q being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnIPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:2
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) )
for p, q, r being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnIPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:3
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) )
for p, q being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnIPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:4
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) )
for p, q being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnIPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:5
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) )
for p, q being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnIPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:6
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) )
for p, q being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnIPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:7
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) )
for q, p being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnIPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:8
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) )
for p, r, q being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnIPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:9
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) )
for p being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds FALSUM : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnIPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:10
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) )
for p, q being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) st p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in CnIPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnIPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds
q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in CnIPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:11
for T, X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) st T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) is IPC_theory & X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) c= T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds
CnIPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) c= T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:12
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) c= CnIPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:13
for X, Y being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) st X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) c= Y : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds
CnIPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) c= CnIPC Y : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:14
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds CnIPC (CnIPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) = CnIPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

registration
let X be ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;
cluster CnIPC X : ( ( ) ( functional ) Element of K6(MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) -> IPC_theory ;
end;

theorem :: INTPRO_1:15
for T being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds
( T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) is IPC_theory iff CnIPC T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) = T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ) ;

theorem :: INTPRO_1:16
for T being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) st T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) is IPC_theory holds
IPC-Taut : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) c= T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

registration
cluster IPC-Taut : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) -> IPC_theory ;
end;

begin

theorem :: INTPRO_1:17
for p being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:18
for q, p being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) st q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) holds
p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:19
IVERUM : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:20
for p, q being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:21
for q, p being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:22
for q, r, p being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:23
for p, q, r being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) st p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) holds
q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:24
for p, q, r being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:25
for p, q, r being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) st p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) holds
(q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:26
for p, q, r being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) st p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) & q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) holds
p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:27
for p, q, r, s being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:28
for p, q, r being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:29
for p, q, r being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:30
for p, q being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:31
for q, p being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => ((q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:32
for s, q, p being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) st s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) & q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) holds
s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

begin

theorem :: INTPRO_1:33
for p being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:34
for p, q being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds
( p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) iff ( p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) & q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: INTPRO_1:35
for p, q being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds
( p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) iff q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ) ;

theorem :: INTPRO_1:36
for p, q, r being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:37
for p, q, r being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:38
for r, p, q being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:39
for p, q being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) '&' p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:40
for p, q, s being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) '&' p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) '&' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:41
for q, s, p being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:42
for q, s, p being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:43
for p, s, q being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:44
for p, q, s being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:45
for p, q, s being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) '&' (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:46
for p, q being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:47
for p, q, s being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) '&' (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:48
for p, q, s being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:49
for p, q, s being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:50
for p, s, q being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' (s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:51
for p, q, s being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) '&' (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:52
for p, q, s being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) '&' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:53
for p, q, s being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) '&' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

begin

theorem :: INTPRO_1:54
for p being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:55
for p, q being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) st ( p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) or q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ) holds
p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:56
for p, q being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:57
for p, q being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds
( p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) iff q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ) ;

theorem :: INTPRO_1:58
for p, q, s being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:59
for p, q, s being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:60
for p, q, s being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:61
for p, q, s being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) st p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) holds
(p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:62
for p, q, s being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:63
for p, q, s being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) st p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) holds
(s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:64
for p, q, s being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:65
for p, q, s being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) 'or' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:66
for p, q, s being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) 'or' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' s : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

begin

definition
let T be ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;
attr T is CPC_theory means :: INTPRO_1:def 19
 for p, q, r being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds
( p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & FALSUM : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => FALSUM : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & ( p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in T : ( ( ) ( ) set ) & p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) implies q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in T : ( ( ) ( ) set ) ) );
end;

theorem :: INTPRO_1:67
for T being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) st T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) is CPC_theory holds
T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) is IPC_theory ;

definition
let X be ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;
func CnCPC X -> ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) means :: INTPRO_1:def 20
 for r being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds
( r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in it : ( ( V15() Function-like ) ( V15() Function-like ) set ) iff for T being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) st T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) is CPC_theory & X : ( ( ) ( ) set ) c= T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds
r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) );
end;

definition
func CPC-Taut -> ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) equals :: INTPRO_1:def 21
 CnCPC ({} MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( empty proper V4() V5() V6() V8() V9() V10() Function-like functional FinSequence-membered ) Element of K6(MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;
end;

theorem :: INTPRO_1:68
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds CnIPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) c= CnCPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:69
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) )
for p, q, r being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds
( p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnCPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) & (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnCPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) & (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnCPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) & (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnCPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnCPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnCPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) & q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnCPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) & (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnCPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) & FALSUM : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnCPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => FALSUM : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnCPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ) ;

theorem :: INTPRO_1:70
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) )
for p, q being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) st p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in CnCPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnCPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds
q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in CnCPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:71
for T, X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) st T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) is CPC_theory & X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) c= T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds
CnCPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) c= T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:72
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) c= CnCPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:73
for X, Y being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) st X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) c= Y : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds
CnCPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) c= CnCPC Y : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:74
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds CnCPC (CnCPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) = CnCPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

registration
let X be ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;
cluster CnCPC X : ( ( ) ( functional ) Element of K6(MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) -> CPC_theory ;
end;

theorem :: INTPRO_1:75
for T being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds
( T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) is CPC_theory iff CnCPC T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional CPC_theory ) Subset of ( ( ) ( non empty ) set ) ) = T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ) ;

theorem :: INTPRO_1:76
for T being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) st T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) is CPC_theory holds
CPC-Taut : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) c= T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

registration
cluster CPC-Taut : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) -> CPC_theory ;
end;

theorem :: INTPRO_1:77
IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) c= CPC-Taut : ( ( ) ( functional CPC_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

begin

definition
let T be ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;
attr T is S4_theory means :: INTPRO_1:def 22
 for p, q, r being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds
( p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & FALSUM : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => FALSUM : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & (Nes (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((Nes p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (Nes q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & (Nes p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & (Nes p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (Nes (Nes p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) & ( p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in T : ( ( ) ( ) set ) & p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) implies q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in T : ( ( ) ( ) set ) ) & ( p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in T : ( ( ) ( ) set ) implies Nes p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in T : ( ( ) ( ) set ) ) );
end;

theorem :: INTPRO_1:78
for T being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) st T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) is S4_theory holds
T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) is CPC_theory ;

theorem :: INTPRO_1:79
for T being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) st T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) is S4_theory holds
T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) is IPC_theory ;

definition
let X be ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;
func CnS4 X -> ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) means :: INTPRO_1:def 23
 for r being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds
( r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in it : ( ( V15() Function-like ) ( V15() Function-like ) set ) iff for T being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) st T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) is S4_theory & X : ( ( ) ( ) set ) c= T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds
r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) );
end;

definition
func S4-Taut -> ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) equals :: INTPRO_1:def 24
 CnS4 ({} MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( empty proper V4() V5() V6() V8() V9() V10() Function-like functional FinSequence-membered ) Element of K6(MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;
end;

theorem :: INTPRO_1:80
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds CnCPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional CPC_theory ) Subset of ( ( ) ( non empty ) set ) ) c= CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:81
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds CnIPC X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) c= CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:82
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) )
for p, q, r being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds
( p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) & (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) & (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) & (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) '&' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) & q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) & (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => r : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) & FALSUM : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) 'or' (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => FALSUM : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ) ;

theorem :: INTPRO_1:83
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) )
for p, q being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) st p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) & p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds
q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:84
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) )
for p, q being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (Nes (p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) => q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => ((Nes p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (Nes q : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:85
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) )
for p being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (Nes p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:86
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) )
for p being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) holds (Nes p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) => (Nes (Nes p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:87
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) )
for p being ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) st p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) in CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds
Nes p : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) Element of MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( V15() V18( NAT : ( ( ) ( non empty V4() V5() V6() ) Element of K6(REAL : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) ) Function-like V32() FinSequence-like FinSubsequence-like ) MC-formula) in CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:88
for T, X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) st T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) is S4_theory & X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) c= T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds
CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) c= T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:89
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) c= CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:90
for X, Y being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) st X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) c= Y : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds
CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) c= CnS4 Y : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:91
for X being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds CnS4 (CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) = CnS4 X : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

registration
let X be ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;
cluster CnS4 X : ( ( ) ( functional ) Element of K6(MC-wff : ( ( ) ( non empty functional with_FALSUM with_int_implication with_int_conjunction with_int_disjunction with_int_propositional_variables with_modal_operator MC-closed ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) -> S4_theory ;
end;

theorem :: INTPRO_1:92
for T being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) holds
( T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) is S4_theory iff CnS4 T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( functional S4_theory ) Subset of ( ( ) ( non empty ) set ) ) = T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ) ;

theorem :: INTPRO_1:93
for T being ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) st T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) is S4_theory holds
S4-Taut : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) c= T : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) ;

registration
cluster S4-Taut : ( ( ) ( functional ) Subset of ( ( ) ( non empty ) set ) ) -> S4_theory ;
end;

theorem :: INTPRO_1:94
CPC-Taut : ( ( ) ( functional CPC_theory ) Subset of ( ( ) ( non empty ) set ) ) c= S4-Taut : ( ( ) ( functional S4_theory ) Subset of ( ( ) ( non empty ) set ) ) ;

theorem :: INTPRO_1:95
IPC-Taut : ( ( ) ( functional IPC_theory ) Subset of ( ( ) ( non empty ) set ) ) c= S4-Taut : ( ( ) ( functional S4_theory ) Subset of ( ( ) ( non empty ) set ) ) ;