:: TOPRNS_1 semantic presentation

begin

definition
let N be ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ;
mode Real_Sequence of N is ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) ) ) sequence of ( ( ) ( ) set ) ) ;
end;

theorem :: TOPRNS_1:1
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) holds
( f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) is ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) iff ( dom f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( ) ( ) set ) = NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) & ( for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) holds f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) is ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) ) ) ) ;

definition
let N be ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let IT be ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;
attr IT is non-zero means :: TOPRNS_1:def 1
 rng IT : ( ( ) ( ) set ) : ( ( ) ( ) set ) c= NonZero (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) Element of K6( the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;
end;

theorem :: TOPRNS_1:2
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for seq being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is non-zero iff for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) holds
seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) . x : ( ( ) ( ) set ) : ( ( ) ( ) set ) <> 0. (TOP-REAL N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V51( TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) ) V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: TOPRNS_1:3
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for seq being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is non-zero iff for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) holds seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) <> 0. (TOP-REAL N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V51( TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) ) V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ;

definition
let N be ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ;
let seq1, seq2 be ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL N : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) ;
func seq1 + seq2 -> ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) ) ) Real_Sequence of N : ( ( ) ( ) 2-sorted ) ) equals :: TOPRNS_1:def 2
seq1 : ( ( ) ( ) set ) + seq2 : ( ( ) ( ) Element of N : ( ( ) ( ) 2-sorted ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) -valued Function-like ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;
end;

definition
let r be ( ( ) ( V11() V12() ext-real ) Real) ;
let N be ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ;
let seq be ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL N : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) ;
func r * seq -> ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( ) set ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL N : ( ( ) ( ) set ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( ) set ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) ) ) Real_Sequence of N : ( ( ) ( ) set ) ) equals :: TOPRNS_1:def 3
r : ( ( ) ( ) 2-sorted ) (#) seq : ( ( ) ( ) Element of r : ( ( ) ( ) 2-sorted ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL N : ( ( ) ( ) set ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) -valued Function-like ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( ) set ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;
end;

definition
let N be ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ;
let seq be ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL N : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) ;
func - seq -> ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) ) ) Real_Sequence of N : ( ( ) ( ) 2-sorted ) ) equals :: TOPRNS_1:def 4
 - seq : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) -valued Function-like ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;
end;

definition
let N be ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ;
let seq1, seq2 be ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL N : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) ;
func seq1 - seq2 -> ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) ) ) Real_Sequence of N : ( ( ) ( ) 2-sorted ) ) equals :: TOPRNS_1:def 5
seq1 : ( ( ) ( ) set ) + (- seq2 : ( ( ) ( ) Element of N : ( ( ) ( ) 2-sorted ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) ) ) Real_Sequence of N : ( ( ) ( ) 2-sorted ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) ) ) Real_Sequence of N : ( ( ) ( ) 2-sorted ) ) ;
end;

definition
let N be ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ;
let seq be ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL N : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) ;
func |.seq.| -> ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) V116() V117() V118() ) Real_Sequence) means :: TOPRNS_1:def 6
 for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) holds it : ( ( ) ( ) Element of N : ( ( ) ( ) 2-sorted ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) = |.(seq : ( ( ) ( ) set ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) ) .| : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) ;
end;

theorem :: TOPRNS_1:4
for N, n being ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat)
for seq1, seq2 being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) holds (seq1 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) + seq2 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) . n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) = (seq1 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) . n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) + (seq2 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) . n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ;

theorem :: TOPRNS_1:5
for r being ( ( ) ( V11() V12() ext-real ) Real)
for N, n being ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat)
for seq being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b2 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b2 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b2 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) holds (r : ( ( ) ( V11() V12() ext-real ) Real) * seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b2 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b2 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b2 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b2 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b2 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b2 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b2 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b2 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) . n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) : ( ( ) ( Relation-like Function-like V39(b2 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b2 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) = r : ( ( ) ( V11() V12() ext-real ) Real) * (seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b2 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b2 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b2 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b2 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) . n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( ) ( Relation-like Function-like V39(b2 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b2 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like V39(b2 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b2 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ;

theorem :: TOPRNS_1:6
for N, n being ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat)
for seq being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) holds (- seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) . n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) = - (seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) . n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real non negative ) Nat) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ;

theorem :: TOPRNS_1:7
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( ) ( V11() V12() ext-real ) Real)
for w being ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) holds (abs r : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) * |.w : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) = |.(r : ( ( ) ( V11() V12() ext-real ) Real) * w : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) ;

theorem :: TOPRNS_1:8
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( ) ( V11() V12() ext-real ) Real)
for seq being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds |.(r : ( ( ) ( V11() V12() ext-real ) Real) * seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) .| : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) V116() V117() V118() ) Real_Sequence) = (abs r : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) (#) |.seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) .| : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) V116() V117() V118() ) Real_Sequence) : ( ( Function-like ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) V116() V117() V118() ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( V116() V117() V118() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: TOPRNS_1:9
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for seq1, seq2 being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds seq1 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) + seq2 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) = seq2 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) + seq1 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: TOPRNS_1:10
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for seq1, seq2, seq3 being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds (seq1 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) + seq2 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) + seq3 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) = seq1 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) + (seq2 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) + seq3 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: TOPRNS_1:11
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for seq being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds - seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) = (- 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V30() V31() ext-real positive non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real non positive ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) * seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: TOPRNS_1:12
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( ) ( V11() V12() ext-real ) Real)
for seq1, seq2 being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds r : ( ( ) ( V11() V12() ext-real ) Real) * (seq1 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) + seq2 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) = (r : ( ( ) ( V11() V12() ext-real ) Real) * seq1 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) + (r : ( ( ) ( V11() V12() ext-real ) Real) * seq2 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: TOPRNS_1:13
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for r, q being ( ( ) ( V11() V12() ext-real ) Real)
for seq being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds (r : ( ( ) ( V11() V12() ext-real ) Real) * q : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) * seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) = r : ( ( ) ( V11() V12() ext-real ) Real) * (q : ( ( ) ( V11() V12() ext-real ) Real) * seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: TOPRNS_1:14
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( ) ( V11() V12() ext-real ) Real)
for seq1, seq2 being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds r : ( ( ) ( V11() V12() ext-real ) Real) * (seq1 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) - seq2 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) = (r : ( ( ) ( V11() V12() ext-real ) Real) * seq1 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) - (r : ( ( ) ( V11() V12() ext-real ) Real) * seq2 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: TOPRNS_1:15
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for seq1, seq2, seq3 being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds seq1 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) - (seq2 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) + seq3 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) = (seq1 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) - seq2 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) - seq3 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: TOPRNS_1:16
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for seq being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V30() V31() ext-real positive non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) * seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) = seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: TOPRNS_1:17
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for seq being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds - (- seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) = seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: TOPRNS_1:18
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for seq1, seq2 being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds seq1 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) - (- seq2 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) = seq1 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) + seq2 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: TOPRNS_1:19
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for seq1, seq2, seq3 being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds seq1 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) - (seq2 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) - seq3 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) = (seq1 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) - seq2 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) + seq3 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: TOPRNS_1:20
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for seq1, seq2, seq3 being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds seq1 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) + (seq2 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) - seq3 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) = (seq1 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) + seq2 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) - seq3 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: TOPRNS_1:21
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( ) ( V11() V12() ext-real ) Real)
for seq being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) st r : ( ( ) ( V11() V12() ext-real ) Real) <> 0 : ( ( ) ( empty V4() V5() V6() V8() V9() V10() V11() V12() V30() V31() ext-real non positive non negative V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) & seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is non-zero holds
r : ( ( ) ( V11() V12() ext-real ) Real) * seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is non-zero ;

theorem :: TOPRNS_1:22
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for seq being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) st seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is non-zero holds
- seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is non-zero ;

theorem :: TOPRNS_1:23
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) holds |.(0. (TOP-REAL N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V51( TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) ) V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) = 0 : ( ( ) ( empty V4() V5() V6() V8() V9() V10() V11() V12() V30() V31() ext-real non positive non negative V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: TOPRNS_1:24
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for w being ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) st |.w : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) = 0 : ( ( ) ( empty V4() V5() V6() V8() V9() V10() V11() V12() V30() V31() ext-real non positive non negative V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) holds
w : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) = 0. (TOP-REAL N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V51( TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) ) V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ;

theorem :: TOPRNS_1:25
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for w being ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) holds |.w : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) >= 0 : ( ( ) ( empty V4() V5() V6() V8() V9() V10() V11() V12() V30() V31() ext-real non positive non negative V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: TOPRNS_1:26
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for w being ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) holds |.(- w : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) = |.w : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) ;

theorem :: TOPRNS_1:27
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for w1, w2 being ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) holds |.(w1 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) - w2 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) = |.(w2 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) - w1 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) ;

theorem :: TOPRNS_1:28
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for w1, w2 being ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) holds
( |.(w1 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) - w2 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) = 0 : ( ( ) ( empty V4() V5() V6() V8() V9() V10() V11() V12() V30() V31() ext-real non positive non negative V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) iff w1 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) = w2 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: TOPRNS_1:29
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for w1, w2 being ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) holds |.(w1 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) + w2 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) <= |.w1 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) + |.w2 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) ;

theorem :: TOPRNS_1:30
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for w1, w2 being ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) holds |.(w1 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) - w2 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) <= |.w1 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) + |.w2 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) ;

theorem :: TOPRNS_1:31
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for w1, w2 being ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) holds |.w1 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) - |.w2 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) <= |.(w1 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) + w2 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) ;

theorem :: TOPRNS_1:32
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for w1, w2 being ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) holds |.w1 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) - |.w2 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) <= |.(w1 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) - w2 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) ;

theorem :: TOPRNS_1:33
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for w1, w2 being ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) st w1 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) <> w2 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) holds
|.(w1 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) - w2 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) > 0 : ( ( ) ( empty V4() V5() V6() V8() V9() V10() V11() V12() V30() V31() ext-real non positive non negative V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: TOPRNS_1:34
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for w1, w2, w being ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) holds |.(w1 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) - w2 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) <= |.(w1 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) - w : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) + |.(w : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) - w2 : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) ;

definition
let N be ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let IT be ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;
attr IT is bounded means :: TOPRNS_1:def 7
 ex r being ( ( ) ( V11() V12() ext-real ) Real) st
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) holds |.(IT : ( ( ) ( ) set ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) ) .| : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) < r : ( ( ) ( V11() V12() ext-real ) Real) ;
end;

theorem :: TOPRNS_1:35
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for seq being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) )
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ex r being ( ( ) ( V11() V12() ext-real ) Real) st
( 0 : ( ( ) ( empty V4() V5() V6() V8() V9() V10() V11() V12() V30() V31() ext-real non positive non negative V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() V12() ext-real ) Real) & ( for m being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) st m : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) holds
|.(seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) . m : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) < r : ( ( ) ( V11() V12() ext-real ) Real) ) ) ;

definition
let N be ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let IT be ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;
attr IT is convergent means :: TOPRNS_1:def 8
 ex g being ( ( ) ( Relation-like Function-like V39(N : ( ( ) ( ) 2-sorted ) ) V40() V116() V117() V118() ) Point of ( ( ) ( ) set ) ) st
for r being ( ( ) ( V11() V12() ext-real ) Real) st 0 : ( ( ) ( empty V4() V5() V6() V8() V9() V10() V11() V12() V30() V31() ext-real non positive non negative V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() V12() ext-real ) Real) holds
ex n being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) st
for m being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) holds
|.((IT : ( ( ) ( ) set ) . m : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) ) - g : ( ( ) ( Relation-like Function-like V39(N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) ) .| : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) < r : ( ( ) ( V11() V12() ext-real ) Real) ;
end;

definition
let N be ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let seq be ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;
assume seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is convergent ;
func lim seq -> ( ( ) ( Relation-like Function-like V39(N : ( ( ) ( ) 2-sorted ) ) V40() V116() V117() V118() ) Point of ( ( ) ( ) set ) ) means :: TOPRNS_1:def 9
 for r being ( ( ) ( V11() V12() ext-real ) Real) st 0 : ( ( ) ( empty V4() V5() V6() V8() V9() V10() V11() V12() V30() V31() ext-real non positive non negative V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() V12() ext-real ) Real) holds
ex n being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) st
for m being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) holds
|.((seq : ( ( ) ( ) set ) . m : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) ) - it : ( ( ) ( ) Element of N : ( ( ) ( ) 2-sorted ) ) ) : ( ( ) ( ) Element of the carrier of (TOP-REAL N : ( ( ) ( ) 2-sorted ) ) : ( ( V195() ) ( V195() ) L16()) : ( ( ) ( ) set ) ) .| : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) < r : ( ( ) ( V11() V12() ext-real ) Real) ;
end;

theorem :: TOPRNS_1:36
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for seq, seq9 being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) st seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is convergent & seq9 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is convergent holds
seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) + seq9 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is convergent ;

theorem :: TOPRNS_1:37
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for seq, seq9 being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) st seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is convergent & seq9 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is convergent holds
lim (seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) + seq9 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) = (lim seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) + (lim seq9 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ;

theorem :: TOPRNS_1:38
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( ) ( V11() V12() ext-real ) Real)
for seq being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) st seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is convergent holds
r : ( ( ) ( V11() V12() ext-real ) Real) * seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is convergent ;

theorem :: TOPRNS_1:39
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( ) ( V11() V12() ext-real ) Real)
for seq being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) st seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is convergent holds
lim (r : ( ( ) ( V11() V12() ext-real ) Real) * seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) = r : ( ( ) ( V11() V12() ext-real ) Real) * (lim seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ;

theorem :: TOPRNS_1:40
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for seq being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) st seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is convergent holds
- seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is convergent ;

theorem :: TOPRNS_1:41
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for seq being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) st seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is convergent holds
lim (- seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) = - (lim seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ;

theorem :: TOPRNS_1:42
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for seq, seq9 being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) st seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is convergent & seq9 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is convergent holds
seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) - seq9 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is convergent ;

theorem :: TOPRNS_1:43
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for seq, seq9 being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) st seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is convergent & seq9 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is convergent holds
lim (seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) - seq9 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) = (lim seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) - (lim seq9 : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ;

theorem :: TOPRNS_1:44
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for seq being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) st seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is convergent holds
seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is bounded ;

theorem :: TOPRNS_1:45
for N being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) )
for seq being ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) st seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) is convergent & lim seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) <> 0. (TOP-REAL N : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V51( TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) ) V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) holds
ex n being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) st
for m being ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) holds
|.(lim seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Point of ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) / 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V30() V31() ext-real positive non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) < |.(seq : ( ( Function-like V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) -valued Function-like total V27( NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) ) Real_Sequence of b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) . m : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like V39(b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) V40() V116() V117() V118() ) Element of the carrier of (TOP-REAL b1 : ( ( ) ( V4() V5() V6() V10() V11() V12() V30() V31() ext-real non negative V126() V127() V128() V129() V130() V131() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V126() V127() V128() V129() V130() V131() V132() ) Element of K6(REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V195() ) ( non empty V70() V138() V139() TopSpace-like V183() V184() V185() V186() V187() V188() V189() V195() ) L16()) : ( ( ) ( non empty ) set ) ) .| : ( ( ) ( V11() V12() ext-real non negative ) Element of REAL : ( ( ) ( non empty V32() V126() V127() V128() V132() ) set ) ) ;