MESFUNC8

:: MESFUNC8 semantic presentation

begin

for X being ( ( non empty ) ( non empty ) set )
for F being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of ( ( ) ( non empty ) Element of bool (bool X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) )
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V44() V46() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) holds
( (superior_setsequence F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) non-ascending ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,(bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V44() V46() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = union (rng (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ^\ n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V44() V46() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,(bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & (inferior_setsequence F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) non-descending ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,(bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V44() V46() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = meet (rng (F : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ^\ n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V44() V46() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,(bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC8:3

theorem :: MESFUNC8:4

theorem :: MESFUNC8:5

theorem :: MESFUNC8:6

for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) )
for F being ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) st F : ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) is convergent & M : ( ( Function-like V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . (Union F : ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) < +infty : ( ( ) ( non empty non real ext-real positive non negative ) set ) holds
ex G being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) st
( G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) = superior_setsequence F : ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) non-ascending ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,(bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) & M : ( ( Function-like V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . (lim F : ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) = inf (rng (M : ( ( Function-like V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) * G : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( non empty countable ) ( non empty V58() countable ) Element of bool ExtREAL : ( ( ) ( non empty V58() ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ;

definition

let X, Y be ( ( ) ( ) set ) ;
let F be ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( ) ( ) set ) ) ;

attr F is with_the_same_dom means :: MESFUNC8:def 1
rng F : ( ( ) ( ) set ) : ( ( ) ( functional ) Element of bool (PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) )) : ( ( ) ( non empty functional ) set ) : ( ( ) ( non empty ) set ) ) is with_common_domain ;

end;

definition

redefine attr F is with_the_same_dom means :: MESFUNC8:def 2
for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds dom (F : ( ( ) ( ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( ) ( ) set ) -valued Function-like ) Element of bool [:X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) = dom (F : ( ( ) ( ) set ) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( ) ( ) set ) -valued Function-like ) Element of bool [:X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ;

end;

registration

let X, Y be ( ( ) ( ) set ) ;

cluster non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom for ( ( ) ( ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;

end;

definition

let X be ( ( non empty ) ( non empty ) set ) ;
let f be ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) ;

func inf f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) means :: MESFUNC8:def 3
( dom it : ( ( ) ( ) set ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) = dom (f : ( ( ) ( ) set ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) & ( for x being ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) st x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) in dom it : ( ( ) ( ) set ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) holds
it : ( ( ) ( ) set ) . x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) = inf (f : ( ( ) ( ) set ) # x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) );

end;

definition

func sup f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) means :: MESFUNC8:def 4
( dom it : ( ( ) ( ) set ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) = dom (f : ( ( ) ( ) set ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) & ( for x being ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) st x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) in dom it : ( ( ) ( ) set ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) holds
it : ( ( ) ( ) set ) . x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) = sup (f : ( ( ) ( ) set ) # x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) );

end;

definition

func inferior_realsequence f -> ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( ) ( ) set ) ) means :: MESFUNC8:def 5
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds
( dom (it : ( ( ) ( ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) = dom (f : ( ( ) ( ) set ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) & ( for x being ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) st x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) in dom (it : ( ( ) ( ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) holds
(it : ( ( ) ( ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) = (inferior_realsequence (f : ( ( ) ( ) set ) # x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) );

end;

definition

func superior_realsequence f -> ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( ) ( ) set ) ) means :: MESFUNC8:def 6
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds
( dom (it : ( ( ) ( ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) = dom (f : ( ( ) ( ) set ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) & ( for x being ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) st x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) in dom (it : ( ( ) ( ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) holds
(it : ( ( ) ( ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) = (superior_realsequence (f : ( ( ) ( ) set ) # x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) );

end;

theorem :: MESFUNC8:7

registration

let X, Y be ( ( ) ( ) set ) ;
let f be ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( ) ( ) set ) ) ;
let n be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V44() V46() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ;

cluster f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ^\ n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V44() V46() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined Function-like total ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total ) set ) -> Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom for ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,Y : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( ) ( ) set ) ) ;

end;

theorem :: MESFUNC8:8

theorem :: MESFUNC8:9

theorem :: MESFUNC8:10

definition

func lim_inf f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) means :: MESFUNC8:def 7
( dom it : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty V57() V58() V59() V60() V61() V62() ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) = dom (f : ( ( ) ( ) set ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) & ( for x being ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) st x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) in dom it : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty V57() V58() V59() V60() V61() V62() ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) holds
it : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real Relation-like Function-like ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) = lim_inf (f : ( ( ) ( ) set ) # x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) );

end;

definition

func lim_sup f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) means :: MESFUNC8:def 8
( dom it : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty V57() V58() V59() V60() V61() V62() ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) = dom (f : ( ( ) ( ) set ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) & ( for x being ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) st x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) in dom it : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty V57() V58() V59() V60() V61() V62() ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) holds
it : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real Relation-like Function-like ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) = lim_sup (f : ( ( ) ( ) set ) # x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) );

end;

theorem :: MESFUNC8:11

for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) holds
( ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) in dom (lim_inf f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
( (lim_inf f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) . x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) = sup (inferior_realsequence (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) & (lim_inf f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) . x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) = sup ((inferior_realsequence f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) & (lim_inf f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) . x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) = (sup (inferior_realsequence f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) . x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ) & lim_inf f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) = sup (inferior_realsequence f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) ) ;

theorem :: MESFUNC8:12

for X being ( ( non empty ) ( non empty ) set )
for f being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) ) holds
( ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) in dom (lim_sup f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
( (lim_sup f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) . x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) = inf (superior_realsequence (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) & (lim_sup f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) . x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) = inf ((superior_realsequence f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) & (lim_sup f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) . x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) = (inf (superior_realsequence f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) . x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ) & lim_sup f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) = inf (superior_realsequence f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) ) ;

theorem :: MESFUNC8:13

definition

func lim f -> ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) means :: MESFUNC8:def 9
( dom it : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty V57() V58() V59() V60() V61() V62() ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) = dom (f : ( ( ) ( ) set ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:X : ( ( ) ( ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) & ( for x being ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) st x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) in dom it : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty V57() V58() V59() V60() V61() V62() ) Element of bool X : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) holds
it : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,(PFuncs (X : ( ( ) ( ) set ) ,f : ( ( ) ( ) set ) )) : ( ( ) ( non empty functional ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real Relation-like Function-like ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) = lim (f : ( ( ) ( ) set ) # x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) );

end;

theorem :: MESFUNC8:14

theorem :: MESFUNC8:15

for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for F being ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds F : ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = (dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (great_dom ((f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ,(R_EAL r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) )) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) holds
union (rng F : ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty ) Element of bool b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) = (dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (great_dom ((sup f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) ,(R_EAL r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) )) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC8:16

for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for F being ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds F : ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = (dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (great_eq_dom ((f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ,(R_EAL r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) )) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) holds
meet (rng F : ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty ) Element of bool b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) = (dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (great_eq_dom ((inf f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) ,(R_EAL r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) )) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC8:17

for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for F being ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds F : ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = (dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (great_dom ((f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ,(R_EAL r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) )) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds (superior_setsequence F : ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) non-ascending ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,(bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = (dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (great_dom (((superior_realsequence f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ,(R_EAL r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) )) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC8:18

for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for F being ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds F : ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = (dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (great_eq_dom ((f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ,(R_EAL r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) )) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds (inferior_setsequence F : ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) non-descending ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,(bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = (dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (great_eq_dom (((inferior_realsequence f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ,(R_EAL r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) )) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC8:19

for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) st dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds (superior_realsequence f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC8:20

for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) st dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds (inferior_realsequence f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC8:21

for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for F being ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds F : ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = (dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (great_eq_dom (((superior_realsequence f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ,(R_EAL r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) )) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) holds
meet F : ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (great_eq_dom ((lim_sup f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) ,(R_EAL r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) )) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC8:22

for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for F being ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) )
for r being ( ( real ) ( V11() real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds F : ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) = (dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (great_dom (((inferior_realsequence f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ,(R_EAL r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) )) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) holds
union (rng F : ( ( b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -valued bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) Element of bool (bool b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ) SetSequence of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( non empty ) Element of bool b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) = (dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) /\ (great_dom ((lim_inf f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) ,(R_EAL r : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) )) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC8:23

for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) st dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) holds
lim_sup f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC8:24

for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) st dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) holds
lim_inf f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC8:25

for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) st dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . n : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) & ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) in E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) holds
f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is convergent ) holds
lim f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC8:26

for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for f being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for g being ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,)
for E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) st dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . n : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) & dom g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) & ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) in E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) holds
( f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is convergent & g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) . x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) = lim (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ) holds
g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is_measurable_on E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: MESFUNC8:27

begin

theorem :: MESFUNC8:28

for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for g being ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,)
for E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) st M : ( ( Function-like V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) < +infty : ( ( ) ( non empty non real ext-real positive non negative ) set ) & dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds
( f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . n : ( ( real ) ( V11() real ext-real ) number ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) & f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . n : ( ( real ) ( V11() real ext-real ) number ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is real-valued ) ) & dom g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) & ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( real ) ( V11() real ext-real ) number ) in E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) holds
( f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) # x : ( ( real ) ( V11() real ext-real ) number ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is convergent_to_finite_number & g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) . x : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) = lim (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) # x : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ) holds
for r, e 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 Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) < r : ( ( real ) ( V11() real ext-real ) number ) & 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) < e : ( ( real ) ( V11() real ext-real ) number ) holds
ex H being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ex N being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st
( H : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) c= E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) & M : ( ( Function-like V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . H : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) < r : ( ( real ) ( V11() real ext-real ) number ) & ( for k being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st N : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) < k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds
for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) in E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) \ H : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) M11(b₁ : ( ( non empty ) ( non empty ) set ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) )) holds
|.(((f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) - (g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) . x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) .| : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) < e : ( ( real ) ( V11() real ext-real ) number ) ) ) ;

theorem :: MESFUNC8:29

for X, Y being ( ( non empty ) ( non empty ) set )
for E being ( ( ) ( ) set )
for F, G being ( ( Function-like V30(b₁ : ( ( non empty ) ( non empty ) set ) ,b₂ : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined b₂ : ( ( non empty ) ( non empty ) set ) -valued Function-like total V30(b₁ : ( ( non empty ) ( non empty ) set ) ,b₂ : ( ( non empty ) ( non empty ) set ) ) ) Function of X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ) set ) ) st ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) holds G : ( ( Function-like V30(b₁ : ( ( non empty ) ( non empty ) set ) ,b₂ : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined b₂ : ( ( non empty ) ( non empty ) set ) -valued Function-like total V30(b₁ : ( ( non empty ) ( non empty ) set ) ,b₂ : ( ( non empty ) ( non empty ) set ) ) ) Function of b₁ : ( ( non empty ) ( non empty ) set ) ,b₂ : ( ( non empty ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of b₂ : ( ( non empty ) ( non empty ) set ) ) = E : ( ( ) ( ) set ) \ (F : ( ( Function-like V30(b₁ : ( ( non empty ) ( non empty ) set ) ,b₂ : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined b₂ : ( ( non empty ) ( non empty ) set ) -valued Function-like total V30(b₁ : ( ( non empty ) ( non empty ) set ) ,b₂ : ( ( non empty ) ( non empty ) set ) ) ) Function of b₁ : ( ( non empty ) ( non empty ) set ) ,b₂ : ( ( non empty ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of b₂ : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b₃ : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) holds
union (rng G : ( ( Function-like V30(b₁ : ( ( non empty ) ( non empty ) set ) ,b₂ : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined b₂ : ( ( non empty ) ( non empty ) set ) -valued Function-like total V30(b₁ : ( ( non empty ) ( non empty ) set ) ,b₂ : ( ( non empty ) ( non empty ) set ) ) ) Function of b₁ : ( ( non empty ) ( non empty ) set ) ,b₂ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Element of bool b₂ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) = E : ( ( ) ( ) set ) \ (meet (rng F : ( ( Function-like V30(b₁ : ( ( non empty ) ( non empty ) set ) ,b₂ : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined b₂ : ( ( non empty ) ( non empty ) set ) -valued Function-like total V30(b₁ : ( ( non empty ) ( non empty ) set ) ,b₂ : ( ( non empty ) ( non empty ) set ) ) ) Function of b₁ : ( ( non empty ) ( non empty ) set ) ,b₂ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Element of bool b₂ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) Element of bool b₃ : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: MESFUNC8:30

for X being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of X : ( ( non empty ) ( non empty ) set ) )
for M being ( ( Function-like V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) )
for f being ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,X : ( ( non empty ) ( non empty ) set ) )
for g being ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,)
for E being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) st dom (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . n : ( ( real ) ( V11() real ext-real ) number ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is_measurable_on E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) & M : ( ( Function-like V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) < +infty : ( ( ) ( non empty non real ext-real positive non negative ) set ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ex L being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) st
( L : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) c= E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) & M : ( ( Function-like V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . (E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) \ L : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) M11(b₁ : ( ( non empty ) ( non empty ) set ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) = 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) & ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( real ) ( V11() real ext-real ) number ) in L : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) holds
|.((f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . n : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) .| : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) < +infty : ( ( ) ( non empty non real ext-real positive non negative ) set ) ) ) ) & ex G being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) st
( G : ( ( real ) ( V11() real ext-real ) number ) c= E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) & M : ( ( Function-like V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . (E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) \ G : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( ) M11(b₁ : ( ( non empty ) ( non empty ) set ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) = 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) & ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) in E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) holds
f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) is convergent_to_finite_number ) & dom g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) : ( ( ) ( ) Element of bool b₁ : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) & ( for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) in G : ( ( real ) ( V11() real ext-real ) number ) holds
g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) . x : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) = lim (f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) # x : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) ) holds
for e 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 Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) < e : ( ( real ) ( V11() real ext-real ) number ) holds
ex F being ( ( ) ( ) Element of S : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) st
( F : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) c= E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) & M : ( ( Function-like V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) zeroed nonnegative sigma-additive ) ( non empty Relation-like b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like total V30(b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) , ExtREAL : ( ( ) ( non empty V58() ) set ) ) ext-real-valued zeroed nonnegative sigma-additive ) sigma_Measure of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) . (E : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) \ F : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) ) : ( ( ) ( ) M11(b₁ : ( ( non empty ) ( non empty ) set ) ,b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) <= e : ( ( real ) ( V11() real ext-real ) number ) & ( for p 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 Relation-like non-zero empty-yielding RAT : ( ( ) ( non empty V33() V57() V58() V59() V60() V63() ) set ) -valued Function-like one-to-one constant functional V44() V46() complex-valued ext-real-valued real-valued natural-valued V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) ) < p : ( ( real ) ( V11() real ext-real ) number ) holds
ex N being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) st N : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) < n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) holds
for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) in F : ( ( ) ( ) Element of b₂ : ( ( non empty compl-closed sigma-multiplicative ) ( non empty compl-closed sigma-multiplicative ) SigmaField of b₁ : ( ( non empty ) ( non empty ) set ) ) ) holds
|.(((f : ( ( Function-like V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) -defined PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) -valued Function-like total V30( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V57() V58() V59() V60() V61() V62() V63() non with_non-empty_elements ) Element of bool REAL : ( ( ) ( non empty V33() V57() V58() V59() V63() non with_non-empty_elements ) set ) : ( ( ) ( non empty ) set ) ) , PFuncs (b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) ) : ( ( ) ( non empty functional ) set ) ) with_the_same_dom ) Functional_Sequence of ( ( ) ( non empty functional ) set ) ,b₁ : ( ( non empty ) ( non empty ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) Element of bool [:b₁ : ( ( non empty ) ( non empty ) set ) ,ExtREAL : ( ( ) ( non empty V58() ) set ) :] : ( ( ) ( non empty Relation-like ext-real-valued ) set ) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) - (g : ( ( Function-like ) ( Relation-like b₁ : ( ( non empty ) ( non empty ) set ) -defined ExtREAL : ( ( ) ( non empty V58() ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) . x : ( ( ) ( ) Element of b₁ : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) .| : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty V58() ) set ) ) < p : ( ( real ) ( V11() real ext-real ) number ) ) ) ;