:: FUZZY_4 semantic presentation

begin

registration
let C1 be ( ( non empty ) ( non empty ) set ) ;
let F be ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(C1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like C1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(C1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Membership_Func of C1 : ( ( non empty ) ( non empty ) set ) ) ;
cluster rng F : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(C1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like C1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(C1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:C1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V43() V44() V45() ) set ) -> non empty ;
end;

theorem :: FUZZY_4:1
for C1 being ( ( non empty ) ( non empty ) set )
for F being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Membership_Func of C1 : ( ( non empty ) ( non empty ) set ) ) holds
( rng F : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Membership_Func of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) is real-bounded & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom F : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Membership_Func of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
F : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Membership_Func of b1 : ( ( non empty ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) <= upper_bound (rng F : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Membership_Func of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in dom F : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Membership_Func of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
F : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Membership_Func of b1 : ( ( non empty ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) >= lower_bound (rng F : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Membership_Func of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ) ;

theorem :: FUZZY_4:2
for C1 being ( ( non empty ) ( non empty ) set )
for F, G being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Membership_Func of C1 : ( ( non empty ) ( non empty ) set ) ) st ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in C1 : ( ( non empty ) ( non empty ) set ) holds
F : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Membership_Func of b1 : ( ( non empty ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) <= G : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Membership_Func of b1 : ( ( non empty ) ( non empty ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) holds
upper_bound (rng F : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Membership_Func of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) <= upper_bound (rng G : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Membership_Func of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ;

theorem :: FUZZY_4:3
for C1, C2 being ( ( non empty ) ( non empty ) set )
for f being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) )
for c being ( ( ) ( ) set ) holds
( 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) <= f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . c : ( ( ) ( ) set ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) & f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . c : ( ( ) ( ) set ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) <= 1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ) ;

definition
let C1, C2 be ( ( non empty ) ( non empty ) set ) ;
let f be ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:C1 : ( ( non empty ) ( non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:C1 : ( ( non empty ) ( non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:C1 : ( ( non empty ) ( non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) ) ;
let x, y be ( ( ) ( ) set ) ;
:: original: .
redefine func f . (x,y) -> ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ;
end;

theorem :: FUZZY_4:4
for C1, C2 being ( ( non empty ) ( non empty ) set )
for f being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) )
for x, y being ( ( ) ( ) set ) holds
( 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) <= f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . (x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) & f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . (x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) <= 1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ) ;

begin

notation
let C1, C2 be ( ( non empty ) ( non empty ) set ) ;
let h be ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:C2 : ( ( non empty ) ( non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:C2 : ( ( non empty ) ( non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:C2 : ( ( non empty ) ( non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) ) ;
synonym converse h for ~ C1;
end;

definition
let C1, C2 be ( ( non empty ) ( non empty ) set ) ;
let h be ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:C2 : ( ( non empty ) ( non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:C2 : ( ( non empty ) ( non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:C2 : ( ( non empty ) ( non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) ) ;
:: original: converse
redefine func converse h -> ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:C1 : ( ( ) ( ) set ) ,C2 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:C1 : ( ( ) ( ) set ) ,C2 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:C1 : ( ( ) ( ) set ) ,C2 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like ) set ) ,C1 : ( ( ) ( ) set ) ) means :: FUZZY_4:def 1
 for x, y being ( ( ) ( ) set ) st [x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) in [:C1 : ( ( ) ( ) set ) ,C2 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) holds
it : ( ( ) ( ) set ) . (x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) = h : ( ( ) ( V28() ) Element of COMPLEX : ( ( ) ( non empty V43() V49() V50() ) set ) ) . (y : ( ( ) ( ) set ) ,x : ( ( ) ( ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ;
end;

theorem :: FUZZY_4:5
for C1, C2 being ( ( non empty ) ( non empty ) set )
for f being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) ) holds converse (converse f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) = f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: FUZZY_4:6
for C1, C2 being ( ( non empty ) ( non empty ) set )
for f being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) ) holds 1_minus (converse f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) = converse (1_minus f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: FUZZY_4:7
for C1, C2 being ( ( non empty ) ( non empty ) set )
for f, g being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) ) holds converse (max (f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ,g : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) = max ((converse f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,(converse g : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: FUZZY_4:8
for C1, C2 being ( ( non empty ) ( non empty ) set )
for f, g being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) ) holds converse (min (f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ,g : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) = min ((converse f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,(converse g : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: FUZZY_4:9
for C1, C2 being ( ( non empty ) ( non empty ) set )
for f, g being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) )
for x, y being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in C1 : ( ( non empty ) ( non empty ) set ) & y : ( ( ) ( ) set ) in C2 : ( ( non empty ) ( non empty ) set ) & f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . [x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) <= g : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . [x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) holds
(converse f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) . [y : ( ( ) ( ) set ) ,x : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) <= (converse g : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) . [y : ( ( ) ( ) set ) ,x : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ;

theorem :: FUZZY_4:10
for C1, C2 being ( ( non empty ) ( non empty ) set )
for f, g being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) ) st g : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) c= holds
converse g : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) c= ;

theorem :: FUZZY_4:11
for C1, C2 being ( ( non empty ) ( non empty ) set )
for f, g being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) ) holds converse (f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) \ g : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) = (converse f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) \ (converse g : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: FUZZY_4:12
for C1, C2 being ( ( non empty ) ( non empty ) set )
for f, g being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) ) holds converse (f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) \+\ g : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) = (converse f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) \+\ (converse g : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) ;

begin

definition
let C1, C2, C3 be ( ( non empty ) ( non empty ) set ) ;
let h be ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:C1 : ( ( non empty ) ( non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:C1 : ( ( non empty ) ( non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:C1 : ( ( non empty ) ( non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) ) ;
let g be ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:C2 : ( ( non empty ) ( non empty ) set ) ,C3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:C2 : ( ( non empty ) ( non empty ) set ) ,C3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:C2 : ( ( non empty ) ( non empty ) set ) ,C3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) ) ;
let x, z be ( ( ) ( ) set ) ;
assume that
x : ( ( ) ( ) set ) in C1 : ( ( non empty ) ( non empty ) set ) and
z : ( ( ) ( ) set ) in C3 : ( ( non empty ) ( non empty ) set ) ;
func min (h,g,x,z) -> ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(C2 : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like C2 : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(C2 : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Membership_Func of C2 : ( ( ) ( ) set ) ) means :: FUZZY_4:def 2
 for y being ( ( ) ( ) Element of C2 : ( ( ) ( ) set ) ) holds it : ( ( ) ( ) Element of bool h : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) . y : ( ( ) ( ) Element of C2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) = min ((h : ( ( ) ( ) set ) . [x : ( ( V6() ) ( Relation-like C3 : ( ( ) ( V28() ) Element of COMPLEX : ( ( ) ( non empty V43() V49() V50() ) set ) ) -defined h : ( ( ) ( ) set ) -valued V6() ) Element of bool [:C3 : ( ( ) ( V28() ) Element of COMPLEX : ( ( ) ( non empty V43() V49() V50() ) set ) ) ,h : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ,y : ( ( ) ( ) Element of C2 : ( ( non empty ) ( non empty ) set ) ) ] : ( ( ) ( ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ,(g : ( ( V6() ) ( Relation-like C1 : ( ( ) ( ) set ) -defined C2 : ( ( ) ( ) set ) -valued V6() ) Element of bool [:C1 : ( ( ) ( ) set ) ,C2 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) . [y : ( ( ) ( ) Element of C2 : ( ( non empty ) ( non empty ) set ) ) ,z : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ;
end;

definition
let C1, C2, C3 be ( ( non empty ) ( non empty ) set ) ;
let h be ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:C1 : ( ( non empty ) ( non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:C1 : ( ( non empty ) ( non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:C1 : ( ( non empty ) ( non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) ) ;
let g be ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:C2 : ( ( non empty ) ( non empty ) set ) ,C3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:C2 : ( ( non empty ) ( non empty ) set ) ,C3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:C2 : ( ( non empty ) ( non empty ) set ) ,C3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) ) ;
func h (#) g -> ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:C1 : ( ( ) ( ) set ) ,C3 : ( ( ) ( V28() ) Element of COMPLEX : ( ( ) ( non empty V43() V49() V50() ) set ) ) :] : ( ( ) ( Relation-like ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:C1 : ( ( ) ( ) set ) ,C3 : ( ( ) ( V28() ) Element of COMPLEX : ( ( ) ( non empty V43() V49() V50() ) set ) ) :] : ( ( ) ( Relation-like ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:C1 : ( ( ) ( ) set ) ,C3 : ( ( ) ( V28() ) Element of COMPLEX : ( ( ) ( non empty V43() V49() V50() ) set ) ) :] : ( ( ) ( Relation-like ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like ) set ) ,C1 : ( ( ) ( ) set ) ) means :: FUZZY_4:def 3
 for x, z being ( ( ) ( ) set ) st [x : ( ( ) ( ) set ) ,z : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) in [:C1 : ( ( ) ( ) set ) ,C3 : ( ( ) ( V28() ) Element of COMPLEX : ( ( ) ( non empty V43() V49() V50() ) set ) ) :] : ( ( ) ( Relation-like ) set ) holds
it : ( ( V6() ) ( Relation-like C3 : ( ( ) ( V28() ) Element of COMPLEX : ( ( ) ( non empty V43() V49() V50() ) set ) ) -defined h : ( ( ) ( ) set ) -valued V6() ) Element of bool [:C3 : ( ( ) ( V28() ) Element of COMPLEX : ( ( ) ( non empty V43() V49() V50() ) set ) ) ,h : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) . (x : ( ( ) ( ) set ) ,z : ( ( ) ( ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) = upper_bound (rng (min (h : ( ( ) ( ) set ) ,g : ( ( V6() ) ( Relation-like C1 : ( ( ) ( ) set ) -defined C2 : ( ( ) ( ) set ) -valued V6() ) Element of bool [:C1 : ( ( ) ( ) set ) ,C2 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ,x : ( ( ) ( ) set ) ,z : ( ( ) ( ) set ) )) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(C2 : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like C2 : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18(C2 : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Membership_Func of C2 : ( ( ) ( ) set ) ) ) : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ;
end;

theorem :: FUZZY_4:13
for C1, C2, C3 being ( ( non empty ) ( non empty ) set )
for f being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) )
for g, h being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) ) holds f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) (#) (max (g : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ,h : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) = max ((f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) (#) g : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ,(f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) (#) h : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: FUZZY_4:14
for C1, C2, C3 being ( ( non empty ) ( non empty ) set )
for f, g being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) )
for h being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) ) holds (max (f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ,g : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) (#) h : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) = max ((f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) (#) h : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ,(g : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) (#) h : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: FUZZY_4:15
for C1, C2, C3 being ( ( non empty ) ( non empty ) set )
for f being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) )
for g, h being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) ) holds min ((f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) (#) g : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ,(f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) (#) h : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) c= ;

theorem :: FUZZY_4:16
for C1, C2, C3 being ( ( non empty ) ( non empty ) set )
for f, g being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) )
for h being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) ) holds min ((f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) (#) h : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ,(g : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) (#) h : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) c= ;

theorem :: FUZZY_4:17
for C1, C2, C3 being ( ( non empty ) ( non empty ) set )
for f being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) )
for g being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) ) holds converse (f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) (#) g : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b3 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b3 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b3 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) = (converse g : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b3 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b3 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b3 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) (#) (converse f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b3 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b3 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b3 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: FUZZY_4:18
for C1, C2, C3 being ( ( non empty ) ( non empty ) set )
for f, g being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) )
for h, k being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) )
for x, z being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in C1 : ( ( non empty ) ( non empty ) set ) & z : ( ( ) ( ) set ) in C3 : ( ( non empty ) ( non empty ) set ) & ( for y being ( ( ) ( ) set ) st y : ( ( ) ( ) set ) in C2 : ( ( non empty ) ( non empty ) set ) holds
( f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . [x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) <= g : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . [x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) & h : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) . [y : ( ( ) ( ) set ) ,z : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) <= k : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) . [y : ( ( ) ( ) set ) ,z : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ) holds
(f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) (#) h : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . [x : ( ( ) ( ) set ) ,z : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) <= (g : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) (#) k : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . [x : ( ( ) ( ) set ) ,z : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ;

theorem :: FUZZY_4:19
for C1, C2, C3 being ( ( non empty ) ( non empty ) set )
for f, g being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) )
for h, k being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) ) st g : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) c= & k : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) c= holds
g : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) (#) k : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) c= ;

begin

definition
let C1, C2 be ( ( non empty ) ( non empty ) set ) ;
func Imf (C1,C2) -> ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:C1 : ( ( ) ( ) set ) ,C2 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:C1 : ( ( ) ( ) set ) ,C2 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:C1 : ( ( ) ( ) set ) ,C2 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like ) set ) ,C1 : ( ( ) ( ) set ) ) means :: FUZZY_4:def 4
 for x, y being ( ( ) ( ) set ) st [x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) in [:C1 : ( ( ) ( ) set ) ,C2 : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) holds
( ( x : ( ( ) ( ) set ) = y : ( ( ) ( ) set ) implies it : ( ( ) ( V28() ) Element of COMPLEX : ( ( ) ( non empty V43() V49() V50() ) set ) ) . (x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) = 1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ) & ( x : ( ( ) ( ) set ) <> y : ( ( ) ( ) set ) implies it : ( ( ) ( V28() ) Element of COMPLEX : ( ( ) ( non empty V43() V49() V50() ) set ) ) . (x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) = 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ) );
end;

theorem :: FUZZY_4:20
for C1, C2 being ( ( non empty ) ( non empty ) set )
for c being ( ( ) ( ) Element of [:C1 : ( ( non empty ) ( non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ) holds
( (Zmf (C1 : ( ( non empty ) ( non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) )) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) . c : ( ( ) ( ) Element of [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) = 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) & (Umf (C1 : ( ( non empty ) ( non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) )) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) . c : ( ( ) ( ) Element of [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) = 1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: FUZZY_4:21
for C1, C2 being ( ( non empty ) ( non empty ) set )
for x, y being ( ( ) ( ) set ) st [x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) in [:C1 : ( ( non empty ) ( non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) holds
( (Zmf (C1 : ( ( non empty ) ( non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) )) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) . [x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) = 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) & (Umf (C1 : ( ( non empty ) ( non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) )) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) . [x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) = 1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: FUZZY_4:22
for C2, C3, C1 being ( ( non empty ) ( non empty ) set )
for f being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) ) holds (Zmf (C1 : ( ( non empty ) ( non empty ) set ) ,C2 : ( ( non empty ) ( non empty ) set ) )) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b3 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b3 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b3 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b3 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) (#) f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b3 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b3 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b3 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) ) = Zmf (C1 : ( ( non empty ) ( non empty ) set ) ,C3 : ( ( non empty ) ( non empty ) set ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b3 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b3 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b3 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b3 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: FUZZY_4:23
for C1, C2, C3 being ( ( non empty ) ( non empty ) set )
for f being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) ) holds f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) (#) (Zmf (C2 : ( ( non empty ) ( non empty ) set ) ,C3 : ( ( non empty ) ( non empty ) set ) )) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b2 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) = Zmf (C1 : ( ( non empty ) ( non empty ) set ) ,C3 : ( ( non empty ) ( non empty ) set ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: FUZZY_4:24
for C1 being ( ( non empty ) ( non empty ) set )
for f being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) ) holds f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) (#) (Zmf (C1 : ( ( non empty ) ( non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) )) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) = (Zmf (C1 : ( ( non empty ) ( non empty ) set ) ,C1 : ( ( non empty ) ( non empty ) set ) )) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) Element of bool [:[:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) ,REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) :] : ( ( ) ( Relation-like non empty V33() V34() V35() ) set ) : ( ( ) ( non empty ) set ) ) (#) f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ;

begin

theorem :: FUZZY_4:25
for X, Y being ( ( non empty ) ( non empty ) set )
for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) )
for y being ( ( ) ( ) Element of Y : ( ( non empty ) ( non empty ) set ) ) holds
( ( x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) = y : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) implies (Imf (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ) set ) )) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . (x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,y : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) = 1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ) & ( x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) <> y : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) implies (Imf (X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ) set ) )) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . (x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,y : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) = 0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ) ) ;

theorem :: FUZZY_4:26
for X, Y being ( ( non empty ) ( non empty ) set )
for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) )
for y being ( ( ) ( ) Element of Y : ( ( non empty ) ( non empty ) set ) )
for f being ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) holds (converse f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b2 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) . (y : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) ,x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) = f : ( ( [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) -defined REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) -valued [.0 : ( ( ) ( Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty V43() V44() V45() V46() V49() V50() ) set ) -valued empty natural V28() real ext-real non positive non negative V33() V34() V35() V36() V43() V44() V45() V46() V47() V48() V49() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) ,1 : ( ( ) ( non empty natural V28() real ext-real positive non negative V43() V44() V45() V46() V47() V48() V60() V61() ) Element of NAT : ( ( ) ( V43() V44() V45() V46() V47() V48() V49() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) ) .] : ( ( ) ( V43() V44() V45() ) Element of bool REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) : ( ( ) ( non empty ) set ) ) -valued V6() V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like non empty ) set ) , REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) V33() V34() V35() ) RMembership_Func of ( ( ) ( Relation-like non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . (x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ,y : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( V28() real ext-real ) Element of REAL : ( ( ) ( non empty V43() V44() V45() V49() V50() ) set ) ) ;