:: PSCOMP_1 semantic presentation

begin

begin

definition
let T be ( ( ) ( ) 1-sorted ) ;
mode RealMap of T is ( ( Function-like V30( the carrier of T : ( ( ) ( ) RLTopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of T : ( ( ) ( ) RLTopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of T : ( ( ) ( ) RLTopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of T : ( ( ) ( ) RLTopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) Function of the carrier of T : ( ( ) ( ) RLTopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ;
end;

registration
let T be ( ( non empty ) ( non empty ) 1-sorted ) ;
cluster non empty Relation-like the carrier of T : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of T : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) ) V30( the carrier of T : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued bounded for ( ( ) ( ) Element of K32(K33( the carrier of T : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty Relation-like complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let T be ( ( ) ( ) 1-sorted ) ;
let f be ( ( Function-like V30( the carrier of T : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of T : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of T : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) V30( the carrier of T : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of T : ( ( ) ( ) 1-sorted ) ) ;
func lower_bound f -> ( ( ) ( V24() real ext-real ) Real) equals :: PSCOMP_1:def 1
 lower_bound (f : ( ( ) ( ) Element of T : ( ( non empty ) ( non empty ) 1-sorted ) ) .: the carrier of T : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V47() V48() V49() ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ;
func upper_bound f -> ( ( ) ( V24() real ext-real ) Real) equals :: PSCOMP_1:def 2
 upper_bound (f : ( ( ) ( ) Element of T : ( ( non empty ) ( non empty ) 1-sorted ) ) .: the carrier of T : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V47() V48() V49() ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ;
end;

theorem :: PSCOMP_1:1
for T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) V44() ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued V44() ) RealMap of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for p being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds f : ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) V44() ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued V44() ) RealMap of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) . p : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) >= lower_bound f : ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) V44() ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued V44() ) RealMap of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( V24() real ext-real ) Real) ;

theorem :: PSCOMP_1:2
for T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) V44() ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued V44() ) RealMap of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for s being ( ( ) ( V24() real ext-real ) Real) st ( for t being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds f : ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) V44() ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued V44() ) RealMap of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) . t : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) >= s : ( ( ) ( V24() real ext-real ) Real) ) holds
lower_bound f : ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) V44() ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued V44() ) RealMap of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( V24() real ext-real ) Real) >= s : ( ( ) ( V24() real ext-real ) Real) ;

theorem :: PSCOMP_1:3
for r being ( ( real ) ( V24() real ext-real ) number )
for T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V30( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st ( for p being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds f : ( ( Function-like V30( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) . p : ( ( real ) ( V24() real ext-real ) number ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) >= r : ( ( real ) ( V24() real ext-real ) number ) ) & ( for t being ( ( real ) ( V24() real ext-real ) number ) st ( for p being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds f : ( ( Function-like V30( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) . p : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) >= t : ( ( real ) ( V24() real ext-real ) number ) ) holds
r : ( ( real ) ( V24() real ext-real ) number ) >= t : ( ( real ) ( V24() real ext-real ) number ) ) holds
r : ( ( real ) ( V24() real ext-real ) number ) = lower_bound f : ( ( Function-like V30( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( V24() real ext-real ) Real) ;

theorem :: PSCOMP_1:4
for T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) V43() ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued V43() ) RealMap of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for p being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds f : ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) V43() ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued V43() ) RealMap of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) . p : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= upper_bound f : ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) V43() ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued V43() ) RealMap of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( V24() real ext-real ) Real) ;

theorem :: PSCOMP_1:5
for T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) V43() ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued V43() ) RealMap of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for t being ( ( real ) ( V24() real ext-real ) number ) st ( for p being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds f : ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) V43() ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued V43() ) RealMap of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) . p : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= t : ( ( real ) ( V24() real ext-real ) number ) ) holds
upper_bound f : ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) V43() ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued V43() ) RealMap of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( V24() real ext-real ) Real) <= t : ( ( real ) ( V24() real ext-real ) number ) ;

theorem :: PSCOMP_1:6
for r being ( ( real ) ( V24() real ext-real ) number )
for T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V30( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st ( for p being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds f : ( ( Function-like V30( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) . p : ( ( real ) ( V24() real ext-real ) number ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= r : ( ( real ) ( V24() real ext-real ) number ) ) & ( for t being ( ( real ) ( V24() real ext-real ) number ) st ( for p being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds f : ( ( Function-like V30( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) . p : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= t : ( ( real ) ( V24() real ext-real ) number ) ) holds
r : ( ( real ) ( V24() real ext-real ) number ) <= t : ( ( real ) ( V24() real ext-real ) number ) ) holds
r : ( ( real ) ( V24() real ext-real ) number ) = upper_bound f : ( ( Function-like V30( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( V24() real ext-real ) Real) ;

theorem :: PSCOMP_1:7
for T being ( ( non empty ) ( non empty ) 1-sorted )
for f being ( ( Function-like V30( the carrier of b1 : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) bounded ) ( non empty Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued bounded_above bounded_below bounded ) RealMap of T : ( ( non empty ) ( non empty ) 1-sorted ) ) holds lower_bound f : ( ( Function-like V30( the carrier of b1 : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) bounded ) ( non empty Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued bounded_above bounded_below bounded ) RealMap of b1 : ( ( non empty ) ( non empty ) 1-sorted ) ) : ( ( ) ( V24() real ext-real ) Real) <= upper_bound f : ( ( Function-like V30( the carrier of b1 : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) bounded ) ( non empty Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued bounded_above bounded_below bounded ) RealMap of b1 : ( ( non empty ) ( non empty ) 1-sorted ) ) : ( ( ) ( V24() real ext-real ) Real) ;

definition
let T be ( ( ) ( ) TopStruct ) ;
let f be ( ( Function-like V30( the carrier of T : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of T : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of T : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of T : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of T : ( ( ) ( ) TopStruct ) ) ;
attr f is continuous means :: PSCOMP_1:def 3
 for Y being ( ( ) ( V47() V48() V49() ) Subset of ( ( ) ( non empty ) set ) ) st Y : ( ( ) ( V47() V48() V49() ) Subset of ( ( ) ( non empty ) set ) ) is closed holds
f : ( ( ) ( ) Element of T : ( ( non empty ) ( non empty ) 1-sorted ) ) " Y : ( ( ) ( V47() V48() V49() ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K32( the carrier of T : ( ( non empty ) ( non empty ) 1-sorted ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) is closed ;
end;

registration
let T be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
cluster non empty Relation-like the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) V30( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous for ( ( ) ( ) Element of K32(K33( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty Relation-like complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( non empty ) set ) ) ;
end;

registration
let T be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let S be ( ( non empty ) ( non empty TopSpace-like ) SubSpace of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ;
cluster non empty Relation-like the carrier of S : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of S : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ) V30( the carrier of S : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous for ( ( ) ( ) Element of K32(K33( the carrier of S : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty Relation-like complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( non empty ) set ) ) ;
end;

theorem :: PSCOMP_1:8
for T being ( ( ) ( ) TopStruct )
for f being ( ( Function-like V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of T : ( ( ) ( ) TopStruct ) ) holds
( f : ( ( Function-like V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b1 : ( ( ) ( ) TopStruct ) ) is continuous iff for Y being ( ( ) ( V47() V48() V49() ) Subset of ( ( ) ( non empty ) set ) ) st Y : ( ( ) ( V47() V48() V49() ) Subset of ( ( ) ( non empty ) set ) ) is open holds
f : ( ( Function-like V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b1 : ( ( ) ( ) TopStruct ) ) " Y : ( ( ) ( V47() V48() V49() ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K32( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) is open ) ;

theorem :: PSCOMP_1:9
for T being ( ( ) ( ) TopStruct )
for f being ( ( Function-like V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of T : ( ( ) ( ) TopStruct ) ) st f : ( ( Function-like V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b1 : ( ( ) ( ) TopStruct ) ) is continuous holds
- f : ( ( Function-like V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b1 : ( ( ) ( ) TopStruct ) ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( Relation-like complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( non empty ) set ) ) is continuous ;

theorem :: PSCOMP_1:10
for r3 being ( ( ) ( V24() real ext-real ) Real)
for T being ( ( ) ( ) TopStruct )
for f being ( ( Function-like V30( the carrier of b2 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of b2 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b2 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of b2 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of T : ( ( ) ( ) TopStruct ) ) st f : ( ( Function-like V30( the carrier of b2 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of b2 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b2 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of b2 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b2 : ( ( ) ( ) TopStruct ) ) is continuous holds
r3 : ( ( ) ( V24() real ext-real ) Real) + f : ( ( Function-like V30( the carrier of b2 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of b2 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b2 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of b2 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b2 : ( ( ) ( ) TopStruct ) ) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b2 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of b2 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33( the carrier of b2 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( Relation-like complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( non empty ) set ) ) is continuous ;

theorem :: PSCOMP_1:11
for T being ( ( ) ( ) TopStruct )
for f being ( ( Function-like V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of T : ( ( ) ( ) TopStruct ) ) st f : ( ( Function-like V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b1 : ( ( ) ( ) TopStruct ) ) is continuous & not 0 : ( ( ) ( empty Relation-like non-empty empty-yielding RAT : ( ( ) ( non empty non trivial V47() V48() V49() V50() V53() V54() non empty-membered ) set ) -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24() real ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued V45() V46() V47() V48() V49() V50() V51() V52() V53() V54() V55() V58() bounded_below bounded_above real-bounded V68() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) in rng f : ( ( Function-like V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b1 : ( ( ) ( ) TopStruct ) ) : ( ( ) ( V47() V48() V49() ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) holds
Inv f : ( ( Function-like V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b1 : ( ( ) ( ) TopStruct ) ) : ( ( Function-like ) ( Relation-like the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( Relation-like complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( non empty ) set ) ) is continuous ;

theorem :: PSCOMP_1:12
for T being ( ( ) ( ) TopStruct )
for f being ( ( Function-like V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of T : ( ( ) ( ) TopStruct ) )
for R being ( ( ) ( ) Subset-Family of ) st f : ( ( Function-like V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b1 : ( ( ) ( ) TopStruct ) ) is continuous & R : ( ( ) ( ) Subset-Family of ) is open holds
(" f : ( ( Function-like V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b1 : ( ( ) ( ) TopStruct ) ) ) : ( ( Function-like V30( bool REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) : ( ( ) ( non empty ) Element of K32(K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of K32(K32( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like bool REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) : ( ( ) ( non empty ) Element of K32(K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of K32(K32( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) -valued Function-like V29( bool REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) : ( ( ) ( non empty ) Element of K32(K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) V30( bool REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) : ( ( ) ( non empty ) Element of K32(K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of K32(K32( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) Element of K32(K33((bool REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) Element of K32(K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ,(bool the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of K32(K32( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty Relation-like ) set ) ) : ( ( ) ( non empty ) set ) ) .: R : ( ( ) ( ) Subset-Family of ) : ( ( ) ( ) Element of K32((bool the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of K32(K32( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) is open ;

theorem :: PSCOMP_1:13
for T being ( ( ) ( ) TopStruct )
for f being ( ( Function-like V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of T : ( ( ) ( ) TopStruct ) )
for R being ( ( ) ( ) Subset-Family of ) st f : ( ( Function-like V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b1 : ( ( ) ( ) TopStruct ) ) is continuous & R : ( ( ) ( ) Subset-Family of ) is closed holds
(" f : ( ( Function-like V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b1 : ( ( ) ( ) TopStruct ) ) ) : ( ( Function-like V30( bool REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) : ( ( ) ( non empty ) Element of K32(K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of K32(K32( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like bool REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) : ( ( ) ( non empty ) Element of K32(K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) -defined bool the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of K32(K32( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) -valued Function-like V29( bool REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) : ( ( ) ( non empty ) Element of K32(K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) V30( bool REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) : ( ( ) ( non empty ) Element of K32(K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) , bool the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) : ( ( ) ( non empty ) Element of K32(K32( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) Element of K32(K33((bool REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) Element of K32(K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ,(bool the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of K32(K32( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty Relation-like ) set ) ) : ( ( ) ( non empty ) set ) ) .: R : ( ( ) ( ) Subset-Family of ) : ( ( ) ( ) Element of K32((bool the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) Element of K32(K32( the carrier of b1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ) is closed ;

definition
let T be ( ( non empty ) ( non empty ) TopStruct ) ;
let f be ( ( Function-like V30( the carrier of T : ( ( non empty ) ( non empty ) TopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of T : ( ( non empty ) ( non empty ) TopStruct ) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of T : ( ( non empty ) ( non empty ) TopStruct ) : ( ( ) ( non empty ) set ) ) V30( the carrier of T : ( ( non empty ) ( non empty ) TopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of T : ( ( non empty ) ( non empty ) TopStruct ) ) ;
let X be ( ( ) ( ) Subset of ) ;
:: original: |
redefine func f | X -> ( ( Function-like V30( the carrier of (T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) | X : ( ( Function-like V30(K33(T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) ) ( Relation-like K33(T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) -defined T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) -valued Function-like V30(K33(T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) ) Element of K32(K33(K33(T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of (T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) | X : ( ( Function-like V30(K33(T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) ) ( Relation-like K33(T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) -defined T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) -valued Function-like V30(K33(T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) ) Element of K32(K33(K33(T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) | X : ( ( Function-like V30(K33(T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) ) ( Relation-like K33(T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) -defined T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) -valued Function-like V30(K33(T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) ) Element of K32(K33(K33(T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V30( the carrier of (T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) | X : ( ( Function-like V30(K33(T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) ) ( Relation-like K33(T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) -defined T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) -valued Function-like V30(K33(T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) ) Element of K32(K33(K33(T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of (T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) | X : ( ( Function-like V30(K33(T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) ) ( Relation-like K33(T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) -defined T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) -valued Function-like V30(K33(T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) ) Element of K32(K33(K33(T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) ) ;
end;

registration
let T be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let f be ( ( Function-like V30( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) continuous ) ( non empty Relation-like the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ;
let X be ( ( ) ( ) Subset of ) ;
cluster f : ( ( Function-like V30( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) continuous ) ( non empty Relation-like the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) V30( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) Element of K32(K33( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty Relation-like complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( non empty ) set ) ) | X : ( ( ) ( ) Element of K32( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( Relation-like ) ( Relation-like X : ( ( ) ( ) Element of K32( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) -defined the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued complex-valued ext-real-valued real-valued ) set ) -> Function-like V30( the carrier of (T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) | X : ( ( ) ( ) Element of K32( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) continuous for ( ( Function-like V30( the carrier of (T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) | X : ( ( ) ( ) Element of K32( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of (T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) | X : ( ( ) ( ) Element of K32( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) | X : ( ( ) ( ) Element of K32( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) ) V30( the carrier of (T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) | X : ( ( ) ( ) Element of K32( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of (T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) | X : ( ( ) ( ) Element of K32( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) ) ;
end;

registration
let T be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let P be ( ( non empty compact ) ( non empty compact ) Subset of ) ;
cluster T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) | P : ( ( non empty compact ) ( non empty compact ) Element of K32( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) -> strict compact ;
end;

begin

theorem :: PSCOMP_1:14
for T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) holds
( ( for f being ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st f : ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is continuous holds
f : ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is with_max ) iff for f being ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st f : ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is continuous holds
f : ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is with_min ) ;

theorem :: PSCOMP_1:15
for T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) holds
( ( for f being ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st f : ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is continuous holds
f : ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is bounded ) iff for f being ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st f : ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is continuous holds
f : ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is with_max ) ;

definition
let T be ( ( ) ( ) TopStruct ) ;
attr T is pseudocompact means :: PSCOMP_1:def 4
 for f being ( ( Function-like V30( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) V30( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) st f : ( ( Function-like V30( the carrier of T : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of T : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of T : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of T : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of T : ( ( ) ( ) TopStruct ) ) is continuous holds
f : ( ( Function-like V30( the carrier of T : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like the carrier of T : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of T : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) ) V30( the carrier of T : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of T : ( ( ) ( ) TopStruct ) ) is bounded ;
end;

registration
cluster non empty TopSpace-like compact -> non empty TopSpace-like pseudocompact for ( ( ) ( ) TopStruct ) ;
end;

registration
cluster non empty TopSpace-like compact for ( ( ) ( ) TopStruct ) ;
end;

registration
let T be ( ( non empty TopSpace-like pseudocompact ) ( non empty TopSpace-like pseudocompact ) TopSpace) ;
cluster Function-like V30( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) continuous -> Function-like V30( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) with_max with_min bounded for ( ( ) ( ) Element of K32(K33( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ,REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty Relation-like complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( non empty ) set ) ) ;
end;

theorem :: PSCOMP_1:16
for T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X being ( ( non empty ) ( non empty ) Subset of )
for Y being ( ( compact ) ( compact ) Subset of )
for f being ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) continuous ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X : ( ( non empty ) ( non empty ) Subset of ) c= Y : ( ( compact ) ( compact ) Subset of ) holds
lower_bound (f : ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) continuous ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) | Y : ( ( compact ) ( compact ) Subset of ) ) : ( ( Function-like V30( the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b3 : ( ( compact ) ( compact ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like b3 : ( ( compact ) ( compact ) Subset of ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b3 : ( ( compact ) ( compact ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b3 : ( ( compact ) ( compact ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( ) set ) ) V30( the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b3 : ( ( compact ) ( compact ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b3 : ( ( compact ) ( compact ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( ) ( V24() real ext-real ) Real) <= lower_bound (f : ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) continuous ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) | X : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( Function-like V30( the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b2 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like b2 : ( ( non empty ) ( non empty ) Subset of ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b2 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b2 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) V30( the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b2 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b2 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( ) ( V24() real ext-real ) Real) ;

theorem :: PSCOMP_1:17
for T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X being ( ( non empty ) ( non empty ) Subset of )
for Y being ( ( compact ) ( compact ) Subset of )
for f being ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) continuous ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X : ( ( non empty ) ( non empty ) Subset of ) c= Y : ( ( compact ) ( compact ) Subset of ) holds
upper_bound (f : ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) continuous ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) | X : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( Function-like V30( the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b2 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like b2 : ( ( non empty ) ( non empty ) Subset of ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b2 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b2 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) V30( the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b2 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b2 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( ) ( V24() real ext-real ) Real) <= upper_bound (f : ( ( Function-like V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) continuous ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) V30( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) | Y : ( ( compact ) ( compact ) Subset of ) ) : ( ( Function-like V30( the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b3 : ( ( compact ) ( compact ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like b3 : ( ( compact ) ( compact ) Subset of ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b3 : ( ( compact ) ( compact ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b3 : ( ( compact ) ( compact ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( ) set ) ) V30( the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b3 : ( ( compact ) ( compact ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) | b3 : ( ( compact ) ( compact ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( ) ( V24() real ext-real ) Real) ;

begin

registration
let n be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real non negative V45() V46() V47() V48() V49() V50() V51() V52() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;
let X, Y be ( ( compact ) ( closed compact ) Subset of ) ;
cluster X : ( ( compact ) ( closed compact ) Element of K32( the carrier of (TOP-REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real non negative V45() V46() V47() V48() V49() V50() V51() V52() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) /\ Y : ( ( compact ) ( closed compact ) Element of K32( the carrier of (TOP-REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real non negative V45() V46() V47() V48() V49() V50() V51() V52() bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) -> compact for ( ( ) ( ) Subset of ) ;
end;

definition
func proj1 -> ( ( Function-like V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) means :: PSCOMP_1:def 5
 for p being ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) holds it : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) . p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ;
func proj2 -> ( ( Function-like V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) means :: PSCOMP_1:def 6
 for p being ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) holds it : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) . p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ;
end;

theorem :: PSCOMP_1:18
for r, s being ( ( real ) ( V24() real ext-real ) number ) holds proj1 : ( ( Function-like V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) " ].r : ( ( real ) ( V24() real ext-real ) number ) ,s : ( ( real ) ( V24() real ext-real ) number ) .[ : ( ( ) ( V47() V48() V49() non left_end non right_end V68() open ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) = { |[r1 : ( ( ) ( V24() real ext-real ) Real) ,r2 : ( ( ) ( V24() real ext-real ) Real) ]| : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) where r1, r2 is ( ( ) ( V24() real ext-real ) Real) : ( r : ( ( real ) ( V24() real ext-real ) number ) < r1 : ( ( ) ( V24() real ext-real ) Real) & r1 : ( ( ) ( V24() real ext-real ) Real) < s : ( ( real ) ( V24() real ext-real ) number ) ) } ;

theorem :: PSCOMP_1:19
for P being ( ( ) ( ) Subset of )
for r3, q3 being ( ( ) ( V24() real ext-real ) Real) st P : ( ( ) ( ) Subset of ) = { |[r1 : ( ( ) ( V24() real ext-real ) Real) ,r2 : ( ( ) ( V24() real ext-real ) Real) ]| : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) where r1, r2 is ( ( ) ( V24() real ext-real ) Real) : ( r3 : ( ( ) ( V24() real ext-real ) Real) < r1 : ( ( ) ( V24() real ext-real ) Real) & r1 : ( ( ) ( V24() real ext-real ) Real) < q3 : ( ( ) ( V24() real ext-real ) Real) ) } holds
P : ( ( ) ( ) Subset of ) is open ;

theorem :: PSCOMP_1:20
for r, s being ( ( real ) ( V24() real ext-real ) number ) holds proj2 : ( ( Function-like V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) " ].r : ( ( real ) ( V24() real ext-real ) number ) ,s : ( ( real ) ( V24() real ext-real ) number ) .[ : ( ( ) ( V47() V48() V49() non left_end non right_end V68() open ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) = { |[r1 : ( ( ) ( V24() real ext-real ) Real) ,r2 : ( ( ) ( V24() real ext-real ) Real) ]| : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) where r1, r2 is ( ( ) ( V24() real ext-real ) Real) : ( r : ( ( real ) ( V24() real ext-real ) number ) < r2 : ( ( ) ( V24() real ext-real ) Real) & r2 : ( ( ) ( V24() real ext-real ) Real) < s : ( ( real ) ( V24() real ext-real ) number ) ) } ;

theorem :: PSCOMP_1:21
for P being ( ( ) ( ) Subset of )
for r3, q3 being ( ( ) ( V24() real ext-real ) Real) st P : ( ( ) ( ) Subset of ) = { |[r1 : ( ( ) ( V24() real ext-real ) Real) ,r2 : ( ( ) ( V24() real ext-real ) Real) ]| : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) where r1, r2 is ( ( ) ( V24() real ext-real ) Real) : ( r3 : ( ( ) ( V24() real ext-real ) Real) < r2 : ( ( ) ( V24() real ext-real ) Real) & r2 : ( ( ) ( V24() real ext-real ) Real) < q3 : ( ( ) ( V24() real ext-real ) Real) ) } holds
P : ( ( ) ( ) Subset of ) is open ;

registration
cluster proj1 : ( ( Function-like V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) -> Function-like V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) continuous ;
cluster proj2 : ( ( Function-like V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) -> Function-like V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) continuous ;
end;

theorem :: PSCOMP_1:22
for X being ( ( ) ( ) Subset of )
for p being ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) st p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) in X : ( ( ) ( ) Subset of ) holds
(proj1 : ( ( Function-like V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) | X : ( ( ) ( ) Subset of ) ) : ( ( Function-like V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | b1 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) Subset of ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | b1 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | b1 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) ) V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | b1 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | b1 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) ) . p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ;

theorem :: PSCOMP_1:23
for X being ( ( ) ( ) Subset of )
for p being ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) st p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) in X : ( ( ) ( ) Subset of ) holds
(proj2 : ( ( Function-like V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) | X : ( ( ) ( ) Subset of ) ) : ( ( Function-like V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | b1 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) Subset of ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | b1 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | b1 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) ) V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | b1 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | b1 : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) ) . p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ;

definition
let X be ( ( ) ( ) Subset of ) ;
func W-bound X -> ( ( ) ( V24() real ext-real ) Real) equals :: PSCOMP_1:def 7
 lower_bound (proj1 : ( ( Function-like V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) | X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( Function-like V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) ) V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) ) : ( ( ) ( V24() real ext-real ) Real) ;
func N-bound X -> ( ( ) ( V24() real ext-real ) Real) equals :: PSCOMP_1:def 8
 upper_bound (proj2 : ( ( Function-like V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) | X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( Function-like V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) ) V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) ) : ( ( ) ( V24() real ext-real ) Real) ;
func E-bound X -> ( ( ) ( V24() real ext-real ) Real) equals :: PSCOMP_1:def 9
 upper_bound (proj1 : ( ( Function-like V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) | X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( Function-like V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) ) V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) ) : ( ( ) ( V24() real ext-real ) Real) ;
func S-bound X -> ( ( ) ( V24() real ext-real ) Real) equals :: PSCOMP_1:def 10
 lower_bound (proj2 : ( ( Function-like V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) | X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( Function-like V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) ) V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued ) RealMap of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) ) : ( ( ) ( V24() real ext-real ) Real) ;
end;

theorem :: PSCOMP_1:24
for p being ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) )
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) st p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) in X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) holds
( W-bound X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( V24() real ext-real ) Real) <= p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= E-bound X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( V24() real ext-real ) Real) & S-bound X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( V24() real ext-real ) Real) <= p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= N-bound X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( V24() real ext-real ) Real) ) ;

definition
let X be ( ( ) ( ) Subset of ) ;
func SW-corner X -> ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) equals :: PSCOMP_1:def 11
|[(W-bound X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( V24() real ext-real ) Real) ,(S-bound X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( V24() real ext-real ) Real) ]| : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;
func NW-corner X -> ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) equals :: PSCOMP_1:def 12
|[(W-bound X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( V24() real ext-real ) Real) ,(N-bound X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( V24() real ext-real ) Real) ]| : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;
func NE-corner X -> ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) equals :: PSCOMP_1:def 13
|[(E-bound X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( V24() real ext-real ) Real) ,(N-bound X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( V24() real ext-real ) Real) ]| : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;
func SE-corner X -> ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) equals :: PSCOMP_1:def 14
|[(E-bound X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( V24() real ext-real ) Real) ,(S-bound X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( V24() real ext-real ) Real) ]| : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;
end;

theorem :: PSCOMP_1:25
for P being ( ( ) ( ) Subset of ) holds (SW-corner P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (NW-corner P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ;

theorem :: PSCOMP_1:26
for P being ( ( ) ( ) Subset of ) holds (SE-corner P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (NE-corner P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ;

theorem :: PSCOMP_1:27
for P being ( ( ) ( ) Subset of ) holds (NW-corner P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (NE-corner P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ;

theorem :: PSCOMP_1:28
for P being ( ( ) ( ) Subset of ) holds (SW-corner P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (SE-corner P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ;

definition
let X be ( ( ) ( ) Subset of ) ;
func W-most X -> ( ( ) ( ) Subset of ) equals :: PSCOMP_1:def 15
(LSeg ((SW-corner X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ,(NW-corner X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) )) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) /\ X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) ;
func N-most X -> ( ( ) ( ) Subset of ) equals :: PSCOMP_1:def 16
(LSeg ((NW-corner X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ,(NE-corner X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) )) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) /\ X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) ;
func E-most X -> ( ( ) ( ) Subset of ) equals :: PSCOMP_1:def 17
(LSeg ((SE-corner X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ,(NE-corner X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) )) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) /\ X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) ;
func S-most X -> ( ( ) ( ) Subset of ) equals :: PSCOMP_1:def 18
(LSeg ((SW-corner X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ,(SE-corner X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) )) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) /\ X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) ;
end;

registration
let X be ( ( non empty compact ) ( non empty closed compact ) Subset of ) ;
cluster W-most X : ( ( non empty compact ) ( non empty closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Subset of ) -> non empty compact ;
cluster N-most X : ( ( non empty compact ) ( non empty closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Subset of ) -> non empty compact ;
cluster E-most X : ( ( non empty compact ) ( non empty closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Subset of ) -> non empty compact ;
cluster S-most X : ( ( non empty compact ) ( non empty closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Subset of ) -> non empty compact ;
end;

definition
let X be ( ( ) ( ) Subset of ) ;
func W-min X -> ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) equals :: PSCOMP_1:def 19
|[(W-bound X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( V24() real ext-real ) Real) ,(lower_bound (proj2 : ( ( Function-like V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) | (W-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( Function-like V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (W-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like W-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( ) Subset of ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (W-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (W-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) ) V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (W-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (W-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) ) ) : ( ( ) ( V24() real ext-real ) Real) ]| : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;
func W-max X -> ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) equals :: PSCOMP_1:def 20
|[(W-bound X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( V24() real ext-real ) Real) ,(upper_bound (proj2 : ( ( Function-like V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) | (W-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( Function-like V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (W-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like W-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( ) Subset of ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (W-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (W-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) ) V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (W-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (W-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) ) ) : ( ( ) ( V24() real ext-real ) Real) ]| : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;
func N-min X -> ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) equals :: PSCOMP_1:def 21
|[(lower_bound (proj1 : ( ( Function-like V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) | (N-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( Function-like V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (N-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like N-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( ) Subset of ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (N-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (N-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) ) V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (N-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (N-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) ) ) : ( ( ) ( V24() real ext-real ) Real) ,(N-bound X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( V24() real ext-real ) Real) ]| : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;
func N-max X -> ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) equals :: PSCOMP_1:def 22
|[(upper_bound (proj1 : ( ( Function-like V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) | (N-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( Function-like V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (N-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like N-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( ) Subset of ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (N-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (N-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) ) V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (N-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (N-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) ) ) : ( ( ) ( V24() real ext-real ) Real) ,(N-bound X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( V24() real ext-real ) Real) ]| : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;
func E-max X -> ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) equals :: PSCOMP_1:def 23
|[(E-bound X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( V24() real ext-real ) Real) ,(upper_bound (proj2 : ( ( Function-like V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) | (E-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( Function-like V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (E-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like E-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( ) Subset of ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (E-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (E-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) ) V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (E-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (E-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) ) ) : ( ( ) ( V24() real ext-real ) Real) ]| : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;
func E-min X -> ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) equals :: PSCOMP_1:def 24
|[(E-bound X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( V24() real ext-real ) Real) ,(lower_bound (proj2 : ( ( Function-like V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) | (E-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( Function-like V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (E-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like E-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( ) Subset of ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (E-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (E-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) ) V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (E-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (E-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) ) ) : ( ( ) ( V24() real ext-real ) Real) ]| : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;
func S-max X -> ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) equals :: PSCOMP_1:def 25
|[(upper_bound (proj1 : ( ( Function-like V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) | (S-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( Function-like V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (S-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like S-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( ) Subset of ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (S-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (S-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) ) V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (S-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (S-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) ) ) : ( ( ) ( V24() real ext-real ) Real) ,(S-bound X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( V24() real ext-real ) Real) ]| : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;
func S-min X -> ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) equals :: PSCOMP_1:def 26
|[(lower_bound (proj1 : ( ( Function-like V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) | (S-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( Function-like V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (S-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( Relation-like S-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( ) Subset of ) -defined the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (S-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) -defined the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (S-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) ) V30( the carrier of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (S-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) : ( ( ) ( ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of ((TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) | (S-most X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset of ) ) : ( ( strict ) ( strict TopSpace-like T_0 T_1 T_2 ) SubSpace of TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) ) ) : ( ( ) ( V24() real ext-real ) Real) ,(S-bound X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( V24() real ext-real ) Real) ]| : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) ;
end;

theorem :: PSCOMP_1:29
for P being ( ( ) ( ) Subset of ) holds
( (SW-corner P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (W-min P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (SW-corner P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (W-max P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (W-min P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (W-max P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (W-min P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (NW-corner P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (W-max P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (NW-corner P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ;

theorem :: PSCOMP_1:30
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) holds
( (SW-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (W-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (SW-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (W-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (SW-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (NW-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (W-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (W-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (W-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (NW-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (W-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (NW-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ;

theorem :: PSCOMP_1:31
for p being ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) )
for Z being ( ( non empty ) ( non empty ) Subset of ) st p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) in W-most Z : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( ) Subset of ) holds
( p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (W-min Z : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & ( Z : ( ( non empty ) ( non empty ) Subset of ) is compact implies ( (W-min Z : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (W-max Z : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ) ) ;

theorem :: PSCOMP_1:32
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) holds W-most X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( non empty closed compact ) Subset of ) c= LSeg ((W-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ,(W-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: PSCOMP_1:33
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) holds LSeg ((W-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ,(W-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) c= LSeg ((SW-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ,(NW-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: PSCOMP_1:34
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) holds
( W-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) in W-most X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( non empty closed compact ) Subset of ) & W-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) in W-most X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( non empty closed compact ) Subset of ) ) ;

theorem :: PSCOMP_1:35
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) holds
( (LSeg ((SW-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ,(W-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) )) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) /\ X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) = {(W-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty V54() ) set ) & (LSeg ((W-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ,(NW-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) )) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) /\ X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) = {(W-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty V54() ) set ) ) ;

theorem :: PSCOMP_1:36
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) st W-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) = W-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) holds
W-most X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( non empty closed compact ) Subset of ) = {(W-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty V54() ) set ) ;

theorem :: PSCOMP_1:37
for P being ( ( ) ( ) Subset of ) holds
( (NW-corner P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (N-min P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (NW-corner P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (N-max P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (N-min P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (N-max P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (N-min P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (NE-corner P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (N-max P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (NE-corner P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ;

theorem :: PSCOMP_1:38
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) holds
( (NW-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (N-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (NW-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (N-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (NW-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (NE-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (N-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (N-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (N-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (NE-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (N-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (NE-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ;

theorem :: PSCOMP_1:39
for p being ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) )
for Z being ( ( non empty ) ( non empty ) Subset of ) st p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) in N-most Z : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( ) Subset of ) holds
( p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (N-min Z : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & ( Z : ( ( non empty ) ( non empty ) Subset of ) is compact implies ( (N-min Z : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (N-max Z : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ) ) ;

theorem :: PSCOMP_1:40
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) holds N-most X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( non empty closed compact ) Subset of ) c= LSeg ((N-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ,(N-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: PSCOMP_1:41
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) holds LSeg ((N-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ,(N-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) c= LSeg ((NW-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ,(NE-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: PSCOMP_1:42
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) holds
( N-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) in N-most X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( non empty closed compact ) Subset of ) & N-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) in N-most X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( non empty closed compact ) Subset of ) ) ;

theorem :: PSCOMP_1:43
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) holds
( (LSeg ((NW-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ,(N-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) )) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) /\ X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) = {(N-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty V54() ) set ) & (LSeg ((N-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ,(NE-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) )) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) /\ X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) = {(N-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty V54() ) set ) ) ;

theorem :: PSCOMP_1:44
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) st N-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) = N-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) holds
N-most X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( non empty closed compact ) Subset of ) = {(N-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty V54() ) set ) ;

theorem :: PSCOMP_1:45
for P being ( ( ) ( ) Subset of ) holds
( (SE-corner P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (E-min P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (SE-corner P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (E-max P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (E-min P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (E-max P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (E-min P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (NE-corner P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (E-max P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (NE-corner P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ;

theorem :: PSCOMP_1:46
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) holds
( (SE-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (E-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (SE-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (E-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (SE-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (NE-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (E-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (E-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (E-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (NE-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (E-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (NE-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ;

theorem :: PSCOMP_1:47
for p being ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) )
for Z being ( ( non empty ) ( non empty ) Subset of ) st p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) in E-most Z : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( ) Subset of ) holds
( p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (E-min Z : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & ( Z : ( ( non empty ) ( non empty ) Subset of ) is compact implies ( (E-min Z : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (E-max Z : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ) ) ;

theorem :: PSCOMP_1:48
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) holds E-most X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( non empty closed compact ) Subset of ) c= LSeg ((E-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ,(E-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: PSCOMP_1:49
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) holds LSeg ((E-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ,(E-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) c= LSeg ((SE-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ,(NE-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: PSCOMP_1:50
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) holds
( E-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) in E-most X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( non empty closed compact ) Subset of ) & E-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) in E-most X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( non empty closed compact ) Subset of ) ) ;

theorem :: PSCOMP_1:51
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) holds
( (LSeg ((SE-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ,(E-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) )) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) /\ X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) = {(E-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty V54() ) set ) & (LSeg ((E-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ,(NE-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) )) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) /\ X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) = {(E-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty V54() ) set ) ) ;

theorem :: PSCOMP_1:52
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) st E-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) = E-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) holds
E-most X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( non empty closed compact ) Subset of ) = {(E-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty V54() ) set ) ;

theorem :: PSCOMP_1:53
for P being ( ( ) ( ) Subset of ) holds
( (SW-corner P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (S-min P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (SW-corner P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (S-max P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (S-min P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (S-max P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (S-min P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (SE-corner P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (S-max P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (SE-corner P : ( ( ) ( ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ;

theorem :: PSCOMP_1:54
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) holds
( (SW-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (S-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (SW-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (S-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (SW-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (SE-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (S-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (S-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (S-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (SE-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & (S-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (SE-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ;

theorem :: PSCOMP_1:55
for p being ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) )
for Z being ( ( non empty ) ( non empty ) Subset of ) st p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) in S-most Z : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( ) Subset of ) holds
( p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = (S-min Z : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `2 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & ( Z : ( ( non empty ) ( non empty ) Subset of ) is compact implies ( (S-min Z : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) & p : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) <= (S-max Z : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) `1 : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ) ) ;

theorem :: PSCOMP_1:56
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) holds S-most X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( non empty closed compact ) Subset of ) c= LSeg ((S-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ,(S-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: PSCOMP_1:57
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) holds LSeg ((S-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ,(S-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) c= LSeg ((SW-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ,(SE-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: PSCOMP_1:58
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) holds
( S-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) in S-most X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( non empty closed compact ) Subset of ) & S-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) in S-most X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( non empty closed compact ) Subset of ) ) ;

theorem :: PSCOMP_1:59
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) holds
( (LSeg ((SW-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ,(S-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) )) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) /\ X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) = {(S-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty V54() ) set ) & (LSeg ((S-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ,(SE-corner X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) )) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) /\ X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( closed compact ) Element of K32( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( non empty ) set ) ) = {(S-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty V54() ) set ) ) ;

theorem :: PSCOMP_1:60
for X being ( ( non empty compact ) ( non empty closed compact ) Subset of ) st S-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) = S-max X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) holds
S-most X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) : ( ( ) ( non empty closed compact ) Subset of ) = {(S-min X : ( ( non empty compact ) ( non empty closed compact ) Subset of ) ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) } : ( ( ) ( non empty V54() ) set ) ;

theorem :: PSCOMP_1:61
for P being ( ( ) ( ) Subset of ) st W-max P : ( ( ) ( ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) = N-min P : ( ( ) ( ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) holds
W-max P : ( ( ) ( ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) = NW-corner P : ( ( ) ( ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ;

theorem :: PSCOMP_1:62
for P being ( ( ) ( ) Subset of ) st N-max P : ( ( ) ( ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) = E-max P : ( ( ) ( ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) holds
N-max P : ( ( ) ( ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) = NE-corner P : ( ( ) ( ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ;

theorem :: PSCOMP_1:63
for P being ( ( ) ( ) Subset of ) st E-min P : ( ( ) ( ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) = S-max P : ( ( ) ( ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) holds
E-min P : ( ( ) ( ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) = SE-corner P : ( ( ) ( ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ;

theorem :: PSCOMP_1:64
for P being ( ( ) ( ) Subset of ) st S-min P : ( ( ) ( ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) = W-min P : ( ( ) ( ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) holds
S-min P : ( ( ) ( ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) = SW-corner P : ( ( ) ( ) Subset of ) : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Point of ( ( ) ( non empty functional ) set ) ) ;

theorem :: PSCOMP_1:65
for r, s being ( ( real ) ( V24() real ext-real ) number ) holds
( proj2 : ( ( Function-like V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) . |[r : ( ( real ) ( V24() real ext-real ) number ) ,s : ( ( real ) ( V24() real ext-real ) number ) ]| : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = s : ( ( real ) ( V24() real ext-real ) number ) & proj1 : ( ( Function-like V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) ) ( non empty Relation-like the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) -defined REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) -valued Function-like V29( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) V30( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) , REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) complex-valued ext-real-valued real-valued continuous ) RealMap of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) ) . |[r : ( ( real ) ( V24() real ext-real ) number ) ,s : ( ( real ) ( V24() real ext-real ) number ) ]| : ( ( ) ( real-valued V71(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) V72() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V47() V48() V49() V50() V51() V52() left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() left_end bounded_below non with_non-empty_elements ) Element of K32(REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V164() V189() V190() V191() V192() V193() V194() V195() strict ) RLTopStruct ) : ( ( ) ( non empty functional ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial V47() V48() V49() V53() V54() non bounded_below non bounded_above V68() non with_non-empty_elements non empty-membered ) set ) ) = r : ( ( real ) ( V24() real ext-real ) number ) ) ;

theorem :: PSCOMP_1:66
for X being ( ( non empty ) ( non empty ) Subset of )
for Y being ( ( compact ) ( closed compact ) Subset of ) st X : ( ( non empty ) ( non empty ) Subset of ) c= Y : ( ( compact ) ( closed compact ) Subset of ) holds
N-bound X : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( V24() real ext-real ) Real) <= N-bound Y : ( ( compact ) ( closed compact ) Subset of ) : ( ( ) ( V24() real ext-real ) Real) ;

theorem :: PSCOMP_1:67
for X being ( ( non empty ) ( non empty ) Subset of )
for Y being ( ( compact ) ( closed compact ) Subset of ) st X : ( ( non empty ) ( non empty ) Subset of ) c= Y : ( ( compact ) ( closed compact ) Subset of ) holds
E-bound X : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( V24() real ext-real ) Real) <= E-bound Y : ( ( compact ) ( closed compact ) Subset of ) : ( ( ) ( V24() real ext-real ) Real) ;

theorem :: PSCOMP_1:68
for X being ( ( non empty ) ( non empty ) Subset of )
for Y being ( ( compact ) ( closed compact ) Subset of ) st X : ( ( non empty ) ( non empty ) Subset of ) c= Y : ( ( compact ) ( closed compact ) Subset of ) holds
S-bound X : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( V24() real ext-real ) Real) >= S-bound Y : ( ( compact ) ( closed compact ) Subset of ) : ( ( ) ( V24() real ext-real ) Real) ;

theorem :: PSCOMP_1:69
for X being ( ( non empty ) ( non empty ) Subset of )
for Y being ( ( compact ) ( closed compact ) Subset of ) st X : ( ( non empty ) ( non empty ) Subset of ) c= Y : ( ( compact ) ( closed compact ) Subset of ) holds
W-bound X : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( V24() real ext-real ) Real) >= W-bound Y : ( ( compact ) ( closed compact ) Subset of ) : ( ( ) ( V24() real ext-real ) Real) ;