RINFSUP1

:: RINFSUP1 semantic presentation

begin

for s, r, t being ( ( real ) ( V11() real ext-real ) number ) holds
( ( s : ( ( real ) ( V11() real ext-real ) number ) - r : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) set ) < t : ( ( real ) ( V11() real ext-real ) number ) & s : ( ( real ) ( V11() real ext-real ) number ) + r : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) set ) > t : ( ( real ) ( V11() real ext-real ) number ) ) iff abs (t : ( ( real ) ( V11() real ext-real ) number ) - s : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) < r : ( ( real ) ( V11() real ext-real ) number ) ) ;

definition

let seq be ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;

func upper_bound seq -> ( ( ) ( V11() real ext-real ) Real) equals :: RINFSUP1:def 1
upper_bound (rng seq : ( ( V52() ) ( V52() ) set ) ) : ( ( ) ( V52() V53() V54() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ;

end;

definition

func lower_bound seq -> ( ( ) ( V11() real ext-real ) Real) equals :: RINFSUP1:def 2
lower_bound (rng seq : ( ( V52() ) ( V52() ) set ) ) : ( ( ) ( V52() V53() V54() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ;

end;

theorem :: RINFSUP1:2

theorem :: RINFSUP1:3

theorem :: RINFSUP1:4

theorem :: RINFSUP1:5

theorem :: RINFSUP1:6

theorem :: RINFSUP1:7

for r being ( ( real ) ( V11() real ext-real ) number )
for seq being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is V88() holds
( r : ( ( real ) ( V11() real ext-real ) number ) = upper_bound seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Real) iff ( ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) holds seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) <= r : ( ( real ) ( V11() real ext-real ) number ) ) & ( for s being ( ( real ) ( V11() real ext-real ) number ) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V52() V53() V54() V55() V56() V57() V58() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( real ) ( V11() real ext-real ) number ) holds
ex k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) st r : ( ( real ) ( V11() real ext-real ) number ) - s : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) set ) < seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ) ) ;

theorem :: RINFSUP1:8

for r being ( ( real ) ( V11() real ext-real ) number )
for seq being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is V89() holds
( r : ( ( real ) ( V11() real ext-real ) number ) = lower_bound seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Real) iff ( ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) holds r : ( ( real ) ( V11() real ext-real ) number ) <= seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) & ( for s being ( ( real ) ( V11() real ext-real ) number ) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V52() V53() V54() V55() V56() V57() V58() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( real ) ( V11() real ext-real ) number ) holds
ex k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) < r : ( ( real ) ( V11() real ext-real ) number ) + s : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) set ) ) ) ) ;

theorem :: RINFSUP1:9

theorem :: RINFSUP1:10

theorem :: RINFSUP1:11

theorem :: RINFSUP1:12

theorem :: RINFSUP1:13

theorem :: RINFSUP1:14

theorem :: RINFSUP1:15

theorem :: RINFSUP1:16

notation

let f be ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;
synonym nonnegative f for nonnegative-yielding ;

end;

definition

redefine attr f is nonnegative-yielding means :: RINFSUP1:def 3
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) holds f : ( ( V52() ) ( V52() ) set ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) >= 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V52() V53() V54() V55() V56() V57() V58() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ;

end;

theorem :: RINFSUP1:17

for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for seq being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is nonnegative holds
seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ^\ k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) subsequence of b₂ : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) is nonnegative ;

theorem :: RINFSUP1:18

theorem :: RINFSUP1:19

theorem :: RINFSUP1:20

theorem :: RINFSUP1:21

theorem :: RINFSUP1:22

theorem :: RINFSUP1:23

theorem :: RINFSUP1:24

theorem :: RINFSUP1:25

theorem :: RINFSUP1:26

for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for seq being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is V88() holds
seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ^\ k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) subsequence of b₂ : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) is V88() ;

theorem :: RINFSUP1:27

for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for seq being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is V89() holds
seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ^\ k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) subsequence of b₂ : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) is V89() ;

theorem :: RINFSUP1:28

for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for seq being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is bounded holds
seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ^\ k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) subsequence of b₂ : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) is bounded ;

theorem :: RINFSUP1:29

theorem :: RINFSUP1:30

for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for seq being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) holds rng (seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ^\ k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) subsequence of b₂ : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( V52() V53() V54() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) = { (seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) where n is ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) } ;

theorem :: RINFSUP1:31

theorem :: RINFSUP1:32

theorem :: RINFSUP1:33

theorem :: RINFSUP1:34

for seq being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is V48() holds
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for R being ( ( ) ( V52() V53() V54() ) Subset of ( ( ) ( ) set ) ) st R : ( ( ) ( V52() V53() V54() ) Subset of ( ( ) ( ) set ) ) = { (seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) where k is ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) } holds
lower_bound R : ( ( ) ( V52() V53() V54() ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) = seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ;

theorem :: RINFSUP1:35

for seq being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is V49() holds
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for R being ( ( ) ( V52() V53() V54() ) Subset of ( ( ) ( ) set ) ) st R : ( ( ) ( V52() V53() V54() ) Subset of ( ( ) ( ) set ) ) = { (seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) where k is ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) } holds
upper_bound R : ( ( ) ( V52() V53() V54() ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) = seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ;

definition

func inferior_realsequence seq -> ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) means :: RINFSUP1:def 4
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Y being ( ( ) ( V52() V53() V54() ) Subset of ( ( ) ( ) set ) ) st Y : ( ( ) ( V52() V53() V54() ) Subset of ( ( ) ( ) set ) ) = { (seq : ( ( V52() ) ( V52() ) set ) . k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) where k is ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) } holds
it : ( ( V11() ) ( V11() ) set ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) = lower_bound Y : ( ( ) ( V52() V53() V54() ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ;

end;

definition

func superior_realsequence seq -> ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) means :: RINFSUP1:def 5
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Y being ( ( ) ( V52() V53() V54() ) Subset of ( ( ) ( ) set ) ) st Y : ( ( ) ( V52() V53() V54() ) Subset of ( ( ) ( ) set ) ) = { (seq : ( ( V52() ) ( V52() ) set ) . k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) where k is ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) } holds
it : ( ( V11() ) ( V11() ) set ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) = upper_bound Y : ( ( ) ( V52() V53() V54() ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ;

end;

theorem :: RINFSUP1:36

theorem :: RINFSUP1:37

theorem :: RINFSUP1:38

theorem :: RINFSUP1:39

theorem :: RINFSUP1:40

theorem :: RINFSUP1:41

theorem :: RINFSUP1:42

theorem :: RINFSUP1:43

theorem :: RINFSUP1:44

theorem :: RINFSUP1:45

theorem :: RINFSUP1:46

theorem :: RINFSUP1:47

theorem :: RINFSUP1:48

theorem :: RINFSUP1:49

theorem :: RINFSUP1:50

theorem :: RINFSUP1:51

theorem :: RINFSUP1:52

theorem :: RINFSUP1:53

theorem :: RINFSUP1:54

theorem :: RINFSUP1:55

for seq being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is bounded holds
upper_bound (inferior_realsequence seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Real) <= lower_bound (superior_realsequence seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: RINFSUP1:56

theorem :: RINFSUP1:57

for seq being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is bounded holds
( inferior_realsequence seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim (inferior_realsequence seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) = upper_bound (inferior_realsequence seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Real) ) ;

theorem :: RINFSUP1:58

for seq being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is bounded holds
( superior_realsequence seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim (superior_realsequence seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) = lower_bound (superior_realsequence seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Real) ) ;

theorem :: RINFSUP1:59

theorem :: RINFSUP1:60

theorem :: RINFSUP1:61

for seq being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is V89() holds
inferior_realsequence seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) = - (superior_realsequence (- seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( V21() ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: RINFSUP1:62

for seq being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is V88() holds
superior_realsequence seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) = - (inferior_realsequence (- seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( V21() ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: RINFSUP1:63

theorem :: RINFSUP1:64

theorem :: RINFSUP1:65

theorem :: RINFSUP1:66

theorem :: RINFSUP1:67

theorem :: RINFSUP1:68

for seq being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is V48() & seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is V88() holds
lower_bound (superior_realsequence seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Real) = upper_bound seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: RINFSUP1:69

theorem :: RINFSUP1:70

theorem :: RINFSUP1:71

theorem :: RINFSUP1:72

theorem :: RINFSUP1:73

theorem :: RINFSUP1:74

for seq being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is V49() & seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is V89() holds
upper_bound (inferior_realsequence seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Real) = lower_bound seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: RINFSUP1:75

theorem :: RINFSUP1:76

for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for seq1, seq2 being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is V89() & seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is V89() holds
(inferior_realsequence (seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) + seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) >= ((inferior_realsequence seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) + ((inferior_realsequence seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ;

theorem :: RINFSUP1:77

for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for seq1, seq2 being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is V88() & seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is V88() holds
(superior_realsequence (seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) + seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) <= ((superior_realsequence seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) + ((superior_realsequence seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ;

theorem :: RINFSUP1:78

for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for seq1, seq2 being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is V89() & seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is nonnegative & seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is V89() & seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is nonnegative holds
(inferior_realsequence (seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) >= ((inferior_realsequence seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) * ((inferior_realsequence seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ;

theorem :: RINFSUP1:79

for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for seq1, seq2 being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is V89() & seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is nonnegative & seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is V89() & seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is nonnegative holds
(inferior_realsequence (seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) >= ((inferior_realsequence seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) * ((inferior_realsequence seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ;

theorem :: RINFSUP1:80

for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for seq1, seq2 being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is V88() & seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is nonnegative & seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is V88() & seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is nonnegative holds
(superior_realsequence (seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) <= ((superior_realsequence seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) * ((superior_realsequence seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ;

definition

func lim_sup seq -> ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) equals :: RINFSUP1:def 6
lower_bound (superior_realsequence seq : ( ( V52() ) ( V52() ) set ) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Real) ;

end;

definition

func lim_inf seq -> ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) equals :: RINFSUP1:def 7
upper_bound (inferior_realsequence seq : ( ( V52() ) ( V52() ) set ) ) : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Real) ;

end;

theorem :: RINFSUP1:81

for r being ( ( real ) ( V11() real ext-real ) number )
for seq being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is bounded holds
( lim_inf seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) <= r : ( ( real ) ( V11() real ext-real ) number ) iff for s being ( ( real ) ( V11() real ext-real ) number ) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V52() V53() V54() V55() V56() V57() V58() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( real ) ( V11() real ext-real ) number ) holds
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ex k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) + k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) < r : ( ( real ) ( V11() real ext-real ) number ) + s : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) set ) ) ;

theorem :: RINFSUP1:82

for r being ( ( real ) ( V11() real ext-real ) number )
for seq being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is bounded holds
( r : ( ( real ) ( V11() real ext-real ) number ) <= lim_inf seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) iff for s being ( ( real ) ( V11() real ext-real ) number ) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V52() V53() V54() V55() V56() V57() V58() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( real ) ( V11() real ext-real ) number ) holds
ex n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) st
for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) holds r : ( ( real ) ( V11() real ext-real ) number ) - s : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) set ) < seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) + k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ;

theorem :: RINFSUP1:83

for r being ( ( real ) ( V11() real ext-real ) number )
for seq being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is bounded holds
( r : ( ( real ) ( V11() real ext-real ) number ) = lim_inf seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) iff for s being ( ( real ) ( V11() real ext-real ) number ) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V52() V53() V54() V55() V56() V57() V58() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( real ) ( V11() real ext-real ) number ) holds
( ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ex k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) + k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) < r : ( ( real ) ( V11() real ext-real ) number ) + s : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) set ) ) & ex n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) st
for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) holds r : ( ( real ) ( V11() real ext-real ) number ) - s : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) set ) < seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) + k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ) ;

theorem :: RINFSUP1:84

for r being ( ( real ) ( V11() real ext-real ) number )
for seq being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is bounded holds
( r : ( ( real ) ( V11() real ext-real ) number ) <= lim_sup seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) iff for s being ( ( real ) ( V11() real ext-real ) number ) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V52() V53() V54() V55() V56() V57() V58() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( real ) ( V11() real ext-real ) number ) holds
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ex k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) + k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) > r : ( ( real ) ( V11() real ext-real ) number ) - s : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) set ) ) ;

theorem :: RINFSUP1:85

for r being ( ( real ) ( V11() real ext-real ) number )
for seq being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is bounded holds
( lim_sup seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) <= r : ( ( real ) ( V11() real ext-real ) number ) iff for s being ( ( real ) ( V11() real ext-real ) number ) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V52() V53() V54() V55() V56() V57() V58() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( real ) ( V11() real ext-real ) number ) holds
ex n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) st
for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) holds seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) + k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) < r : ( ( real ) ( V11() real ext-real ) number ) + s : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) set ) ) ;

theorem :: RINFSUP1:86

for r being ( ( real ) ( V11() real ext-real ) number )
for seq being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is bounded holds
( r : ( ( real ) ( V11() real ext-real ) number ) = lim_sup seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) iff for s being ( ( real ) ( V11() real ext-real ) number ) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V52() V53() V54() V55() V56() V57() V58() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( real ) ( V11() real ext-real ) number ) holds
( ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ex k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) + k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) > r : ( ( real ) ( V11() real ext-real ) number ) - s : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) set ) ) & ex n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) st
for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) holds seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) + k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real V52() V53() V54() V55() V56() V57() V69() V70() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) < r : ( ( real ) ( V11() real ext-real ) number ) + s : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) set ) ) ) ;

theorem :: RINFSUP1:87

theorem :: RINFSUP1:88

theorem :: RINFSUP1:89

for seq being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent holds
( lim seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) = lim_sup seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) & lim seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) = lim_inf seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ;

theorem :: RINFSUP1:90

for seq being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is bounded holds
( lim_sup (- seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) = - (lim_inf seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) & lim_inf (- seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) = - (lim_sup seq : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ;

theorem :: RINFSUP1:91

theorem :: RINFSUP1:92

for seq1, seq2 being ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is bounded & seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is bounded holds
( (lim_inf seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) + (lim_inf seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) <= lim_inf (seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) + seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) & lim_inf (seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) + seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) <= (lim_inf seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) + (lim_sup seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) & lim_inf (seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) + seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) <= (lim_sup seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) + (lim_inf seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) & (lim_inf seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) + (lim_sup seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) <= lim_sup (seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) + seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) & (lim_sup seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) + (lim_inf seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) <= lim_sup (seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) + seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) & lim_sup (seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) + seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) <= (lim_sup seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) + (lim_sup seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) & ( ( seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent or seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent ) implies ( lim_inf (seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) + seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) = (lim_inf seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) + (lim_inf seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) & lim_sup (seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) + seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) = (lim_sup seq1 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) + (lim_sup seq2 : ( ( V21() V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) V21() V26( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() V90() ) Element of K6(REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V52() V53() V54() V58() V71() V90() ) set ) ) ) ) ) ;

theorem :: RINFSUP1:93